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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, a Banach space (pronounced ) is a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space." Banach spaces originally grew out of the study of
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...
, the spaces under study are often Banach spaces.

Definition

A Banach space is a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...
normed space $\left(X, \, \cdot \, \right).$ A normed space is a pairIt is common to read "$X$ is a normed space" instead of the more technically correct but (usually) pedantic "$\left(X, \, \cdot \, \right)$ is a normed space," especially if the norm is well known (for example, such as with $L^p$ spaces) or when there is no particular need to choose any one (equivalent) norm over any other (especially in the more abstract theory of
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s), in which case this norm (if needed) is often automatically assumed to be denoted by $\, \cdot \, .$ However, in situations where emphasis is placed on the norm, it is common to see $\left(X, \, \cdot \, \right)$ written instead of $X.$ The technically correct definition of normed spaces as pairs $\left(X, \, \cdot \, \right)$ may also become important in the context of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
where the distinction between the categories of normed spaces,
normable space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
s,
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
s,
TVS TVS may refer to: Mathematics * Topological vector space Television * Television Sydney, TV channel in Sydney, Australia * Television South, ITV franchise holder in the South of England between 1982 and 1992 * TVS Television Network, US dist ...
s,
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
s, etc. is usually important.
$\left(X, \, \cdot \, \right)$ consisting of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
$X$ over a scalar field $\mathbb$ (where $\mathbb$ is commonly $\R$ or $\Complex$) together with a distinguishedThis means that if the norm $\, \cdot \,$ is replaced with a different norm $\, \,\cdot\,\, ^ \text X,$ then $\left(X, \, \cdot \, \right)$ is the same normed space as $\left\left(X, \, \cdot \, ^\right\right),$ even if the norms are equivalent. However, equivalence of norms on a given vector space does form an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
.
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
$\, \cdot \, : X \to \R.$ Like all norms, this norm induces a
translation invariant In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...
A metric $D$ on a vector space $X$ is said to be translation invariant if $D\left(x, y\right) = D\left(x + z, y + z\right)$ for all vectors $x, y, z \in X.$ This happens if and only if $D\left(x, y\right) = D\left(x - y, 0\right)$ for all vectors $x, y \in X.$ A metric that is induced by a norm is always translation invariant. distance function, called the canonical or (norm) induced metric, defined byBecause $\, - z\, = \, z\,$ for all $z \in X,$ it is always true that $d\left(x, y\right) := \, y - x\, = \, x - y\,$ for all $x, y \in X.$ So the order of $x$ and $y$ in this definition does not matter. $d(x, y) := \, y - x\, = \, x - y\,$ for all vectors $x, y \in X.$ This makes $X$ into a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
$\left(X, d\right).$ A sequence $x_ = \left\left(x_n\right\right)_^$ is called or or if for every real $r > 0,$ there exists some index $N$ such that $d\left(x_n, x_m\right) = \left\, x_n - x_m\right\, < r$ whenever $m$ and $n$ are greater than $N.$ The canonical metric $d$ is called a if the pair $\left(X, d\right)$ is a , which by definition means for every $x_ = \left\left(x_n\right\right)_^$ in $\left(X, d\right),$ there exists some $x \in X$ such that $\lim_ \left\, x_n - x\right\, = 0$ where because $\left\, x_n - x\right\, = d\left\left(x_n, x\right\right),$ this sequence's convergence to $x$ can equivalently be expressed as: $\lim_ x_n = x \; \text (X, d).$ By definition, the normed space $\left(X, \, \cdot \, \right)$ is a if the norm induced metric $d$ is a
complete metric In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
, or said differently, if $\left(X, d\right)$ is a complete metric space. The norm $\, \cdot \,$ of a normed space $\left(X, \, \cdot \, \right)$ is called a if $\left(X, \, \cdot \, \right)$ is a Banach space. L-semi-inner product For any normed space $\left(X, \, \cdot \, \right),$ there exists an
L-semi-inner product In mathematics, there are two different notions of semi-inner-product. The first, and more common, is that of an inner product which is not required to be strictly positive. This article will deal with the second, called a L-semi-inner product or s ...
$\langle \cdot, \cdot \rangle$ on $X$ such that $\, x\, = \sqrt$ for all $x \in X$; in general, there may be infinitely many L-semi-inner products that satisfy this condition. L-semi-inner products are a generalization of inner products, which are what fundamentally distinguish
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
s from all other Banach spaces. This shows that all normed spaces (and hence all Banach spaces) can be considered as being generalizations of (pre-)Hilbert spaces. Characterization in terms of series The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space $X$ is a Banach space if and only if each
absolutely convergent In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is sai ...
series in $X$ converges in $X,$ $\sum_^ \, v_n\, < \infty \quad \text \quad \sum_^ v_n\ \ \text \ \ X.$

Topology

The canonical metric $d$ of a normed space $\left(X, \, \cdot\, \right)$ induces the usual metric topology $\tau_d$ on $X,$ which is referred to as the canonical or norm induced
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. Every normed space is automatically assumed to carry this Hausdorff topology, unless indicated otherwise. With this topology, every Banach space is a Baire space, although there exist normed spaces that are Baire but not Banach. The norm $\, \,\cdot\,\, : \left\left(X, \tau_d\right\right) \to \R$ is always a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
with respect to the topology that it induces. The open and closed balls of radius $r > 0$ centered at a point $x \in X$ are, respectively, the sets $B_r(x) := \ \qquad \text \qquad C_r(x) := \.$ Any such ball is a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
and bounded subset of $X,$ but a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
ball/
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
exists if and only if $X$ is a finite-dimensional vector space. In particular, no infinite–dimensional normed space can be locally compact or have the Heine–Borel property. If $x_0$ is a vector and $s \neq 0$ is a scalar then $x_0 + s B_r(x) = B_\left(x_0 + s x\right) \qquad \text \qquad x_0 + s C_r(x) = C_\left(x_0 + s x\right).$ Using $s := 1$ shows that this norm-induced topology is
translation invariant In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...
, which means that for any $x \in X$ and $S \subseteq X,$ the subset $S$ is
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (Y ...
(respectively, closed) in $X$ if and only if this is true of its translation $x + S := \.$ Consequently, the norm induced topology is completely determined by any neighbourhood basis at the origin. Some common neighborhood bases at the origin include: $\left\, \qquad \left\, \qquad \left\, \qquad \text \qquad \left\$ where $\left\left(r_n\right\right)_^$ is a sequence in of positive real numbers that converges to $0$ in $\R$ (such as $r_n := 1/n$ or $r_n := 1/2^n$ for instance). So for example, every open subset $U$ of $X$ can be written as a union $U = \bigcup_ B_(x) = \bigcup_ x + B_(0) = \bigcup_ x + r_x B_1(0)$ indexed by some subset $I \subseteq U,$ where every $r_x$ is of the form $r_x = \tfrac$ for some integer $n_x > 0$ (the closed ball can also be used instead of the open ball, although the indexing set $I$ and radii $r_x$ may need to be changed). Additionally, $I$ can always be chosen to be
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
if $X$ is a , which by definition means that $X$ contains some countable
dense subset In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
. The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space is homeomorphic to the product space $\prod_ \R$ of countably many copies of $\R$ (this homeomorphism need not be a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
). Since every Banach space is a Fréchet space, this is also true of all infinite–dimensional separable Banach spaces, including the separable Hilbert $\ell$2 sequence space $\ell^2\left(\N\right)$ with its usual norm $\, \cdot\, _2,$ where (in sharp contrast to finite−dimensional spaces) $\ell^2\left(\N\right)$ is also homeomorphic to its unit $\left\.$ There is a compact subset $S$ of $\ell^2\left(\N\right)$ whose
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean spac ...
$\operatorname\left(S\right)$ is closed and thus also compact (see this footnoteLet $H$ be the separable
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
$\ell^2\left(\N\right)$ of square-summable sequences with the usual norm $\, \cdot\, _2$ and let $e_n = \left(0, \ldots, 0, 1, 0, \ldots\right)$ be the standard orthonormal basis (that is $1$ at the $n^$-coordinate). The closed set $S = \ \cup \left\$ is compact (because it is sequentially compact) but its convex hull $\operatorname S$ is a closed set because $h := \sum_^ \tfrac \tfrac e_n$ belongs to the closure of $\operatorname S$ in $H$ but $h \not\in\operatorname S$ (since every sequence $\left\left(z_n\right\right)_^\infty \in \operatorname S$ is a finite
convex combination In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other w ...
of elements of $S$ and so $z_n = 0$ for all but finitely many coordinates, which is not true of $h$). However, like in all
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...
Hausdorff locally convex spaces, the convex hull $K := \overline S$ of this compact subset is compact. The vector subspace $X := \operatorname S = \operatorname \left\$ is a pre-Hilbert space when endowed with the substructure that the Hilbert space $H$ induces on it but $X$ is not complete and $h \not\in C := K \cap X$ (since $h \not\in X$). The closed convex hull of $S$ in $X$ (here, "closed" means with respect to $X,$ and not to $H$ as before) is equal to $K \cap X,$ which is not compact (because it is not a complete subset). This shows that in a Hausdorff locally convex space that is not complete, the closed convex hull of compact subset might to be compact (although it will be precompact/totally bounded).
for an example). However, like in all Banach spaces, the convex hull $\overline S$ of this (and every other) compact subset will be compact. But if a normed space is not complete then it is in general guaranteed that $\overline S$ will be compact whenever $S$ is; an example can even be found in a (non-complete) pre-Hilbert vector subspace of $\ell^2\left(\N\right).$ This norm-induced topology also makes $\left\left(X, \tau_d\right\right)$ into what is known as a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
(TVS), which by definition is a vector space endowed with a topology making the operations of addition and scalar multiplication continuous. It is emphasized that the TVS $\left\left(X, \tau_d\right\right)$ is a vector space together with a certain type of topology; that is to say, when considered as a TVS, it is associated with particular norm or metric (both of which are " forgotten"). This Hausdorff TVS $\left\left(X, \tau_d\right\right)$ is even locally convex because the set of all open balls centered at the origin forms a neighbourhood basis at the origin consisting of convex
balanced In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
open sets. This TVS is also , which by definition refers to any TVS whose its topology is induced by some (possibly unknown)
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
. Comparison of complete metrizable vector topologies The open mapping theorem implies that if $\tau \text \tau_2$ are topologies on $X$ that make both $\left(X, \tau\right)$ and $\left\left(X, \tau_2\right\right)$ into complete metrizable TVS (for example, Banach or Fréchet spaces) and if one topology is finer or coarser than the other then they must be equal (that is, if $\tau \subseteq \tau_2 \text \tau_2 \subseteq \tau \text \tau = \tau_2$). So for example, if $\left(X, p\right) \text \left(X, q\right)$ are Banach spaces with topologies $\tau_p \text \tau_q$ and if one of these spaces has some open ball that is also an open subset of the other space (or equivalently, if one of $p : \left\left(X, \tau_q\right\right) \to \R$ or $q : \left\left(X, \tau_p\right\right) \to \R$ is continuous) then their topologies are identical and their norms are equivalent.

Completeness

Complete norms and equivalent norms Two norms, $p$ and $q,$ on a vector space are said to be if they induce the same topology; this happens if and only if there exist positive real numbers $c, C > 0$ such that $c q(x) \leq p(x) \leq C q(x)$ for all $x \in X.$ If $p$ and $q$ are two equivalent norms on a vector space $X$ then $\left(X, p\right)$ is a Banach space if and only if $\left(X, q\right)$ is a Banach space. See this footnote for an example of a continuous norm on a Banach space that is equivalent to that Banach space's given norm.Let denote the Banach space of continuous functions with the supremum norm and let $\tau_$ denote the topology on induced by $\, \cdot\, _.$ The vector space can be identified (via the inclusion map) as a proper dense vector subspace $X$ of the $L^1$ space which satisfies $\, f\, _1 \leq \, f\, _$ for all $f \in X.$ Let $p$ denote the restriction of the L1-norm to $X,$ which makes this map $p : X \to \R$ a norm on $X$ (in general, the restriction of any norm to any vector subspace will necessarily again be a norm). The normed space $\left(X, p\right)$ is a Banach space since its completion is the proper superset Because $p \leq \, \cdot\, _$ holds on $X,$ the map $p : \left\left(X, \tau_\right\right) \to \R$ is continuous. Despite this, the norm $p$ is equivalent to the norm $\, \cdot\, _$ (because $\left\left(X, \, \cdot\, _\right\right)$ is complete but $\left(X, p\right)$ is not). All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space. Complete norms vs complete metrics A metric $D$ on a vector space $X$ is induced by a norm on $X$ if and only if $D$ is
translation invariant In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...
and , which means that $D\left(sx, sy\right) = , s, D\left(x, y\right)$ for all scalars $s$ and all $x, y \in X,$ in which case the function $\, x\, := D\left(x, 0\right)$ defines a norm on $X$ and the canonical metric induced by $\, \cdot\,$ is equal to $D.$ Suppose that $\left(X, \, \cdot\, \right)$ is a normed space and that $\tau$ is the norm topology induced on $X.$ Suppose that $D$ is metric on $X$ such that the topology that $D$ induces on $X$ is equal to $\tau.$ If $D$ is
translation invariant In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...
then $\left(X, \, \cdot\, \right)$ is a Banach space if and only if $\left(X, D\right)$ is a complete metric space. If $D$ is translation invariant, then it may be possible for $\left(X, \, \cdot\, \right)$ to be a Banach space but for $\left(X, D\right)$ to be a complete metric space (see this footnoteThe normed space $\left(\R,, \cdot , \right)$ is a Banach space where the absolute value is a
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
on the real line $\R$ that induces the usual Euclidean topology on $\R.$ Define a metric $D : \R \times \R \to \R$ on $\R$ by $D\left(x, y\right) =, \arctan\left(x\right) - \arctan\left(y\right),$ for all $x, y \in \R.$ Just like induced metric, the metric $D$ also induces the usual Euclidean topology on $\R.$ However, $D$ is not a complete metric because the sequence $x_ = \left\left(x_i\right\right)_^$ defined by $x_i := i$ is a sequence but it does not converge to any point of $\R.$ As a consequence of not converging, this sequence cannot be a Cauchy sequence in $\left(\R,, \cdot , \right)$ (that is, it is not a Cauchy sequence with respect to the norm $, \cdot,$) because if it was then the fact that $\left(\R,, \cdot , \right)$ is a Banach space would imply that it converges (a contradiction).
for an example). In contrast, a theorem of Klee,The statement of the theorem is: Let $d$ be metric on a vector space $X$ such that the topology $\tau$ induced by $d$ on $X$ makes $\left(X, \tau\right)$ into a topological vector space. If $\left(X, d\right)$ is a complete metric space then $\left(X, \tau\right)$ is a complete topological vector space. which also applies to all metrizable topological vector spaces, implies that if there exists This metric $D$ is assumed to be translation-invariant. So in particular, this metric $D$ does even have to be induced by a norm. complete metric $D$ on $X$ that induces the norm topology $\tau$ on $X,$ then $\left(X, \, \cdot\, \right)$ is a Banach space. A Fréchet space is a
locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
whose topology is induced by some translation-invariant complete metric. Every Banach space is a Fréchet space but not conversely; indeed, there even exist Fréchet spaces on which no norm is a continuous function (such as the space of real sequences $\R^ = \prod_ \R$ with the product topology). However, the topology of every Fréchet space is induced by some
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
family of real-valued (necessarily continuous) maps called seminorms, which are generalizations of
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
s. It is even possible for a Fréchet space to have a topology that is induced by a countable family of (such norms would necessarily be continuous)A norm (or seminorm) $p$ on a topological vector space $\left(X, \tau\right)$ is continuous if and only if the topology $\tau_p$ that $p$ induces on $X$ is coarser than $\tau$ (meaning, $\tau_p \subseteq \tau$), which happens if and only if there exists some open ball $B$ in $\left(X, p\right)$ (such as maybe $\$ for example) that is open in $\left(X, \tau\right).$ but to not be a Banach/
normable space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
because its topology can not be defined by any norm. An example of such a space is the Fréchet space $C^\left(K\right),$ whose definition can be found in the article on spaces of test functions and distributions. Complete norms vs complete topological vector spaces There is another notion of completeness besides metric completeness and that is the notion of a complete topological vector space (TVS) or TVS-completeness, which uses the theory of
uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uni ...
s. Specifically, the notion of TVS-completeness uses a unique translation-invariant uniformity, called the canonical uniformity, that depends on vector subtraction and the topology $\tau$ that the vector space is endowed with, and so in particular, this notion of TVS completeness is independent of whatever norm induced the topology $\tau$ (and even applies to TVSs that are even metrizable). Every Banach space is a complete TVS. Moreover, a normed space is a Banach space (that is, its norm-induced metric is complete) if and only if it is complete as a topological vector space. If $\left(X, \tau\right)$ is a metrizable topological vector space (such as any norm induced topology, for example), then $\left(X, \tau\right)$ is a complete TVS if and only if it is a complete TVS, meaning that it is enough to check that every Cauchy in $\left(X, \tau\right)$ converges in $\left(X, \tau\right)$ to some point of $X$ (that is, there is no need to consider the more general notion of arbitrary Cauchy nets). If $\left(X, \tau\right)$ is a topological vector space whose topology is induced by (possibly unknown) norm (such spaces are called and they are characterized by being Hausdorff and having a bounded
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
neighborhood of the origin), then $\left(X, \tau\right)$ is a complete topological vector space if and only if $X$ may be assigned a
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
$\, \cdot\,$ that induces on $X$ the topology $\tau$ and also makes $\left(X, \, \cdot\, \right)$ into a Banach space. A Hausdorff
locally convex topological vector space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological v ...
$X$ is normable if and only if its strong dual space $X^_b$ is normable, in which case $X^_b$ is a Banach space ($X^_b$ denotes the strong dual space of $X,$ whose topology is a generalization of the dual norm-induced topology on the continuous dual space $X^$; see this footnote$X^$ denotes the continuous dual space of $X.$ When $X^$ is endowed with the strong dual space topology, also called the topology of uniform convergence on bounded subsets of $X,$ then this is indicated by writing $X^_b$ (sometimes, the subscript $\beta$ is used instead of $b$). When $X$ is a normed space with norm $\, \cdot\,$ then this topology is equal to the topology on $X^$ induced by the dual norm. In this way, the strong topology is a generalization of the usual dual norm-induced topology on $X^.$ for more details). If $X$ is a metrizable locally convex TVS, then $X$ is normable if and only if $X^_b$ is a
Fréchet–Urysohn space In the field of topology, a Fréchet–Urysohn space is a topological space X with the property that for every subset S \subseteq X the closure of S in X is identical to the ''sequential'' closure of S in X. Fréchet–Urysohn spaces are a speci ...
.Gabriyelyan, S.S
"On topological spaces and topological groups with certain local countable networks
(2014)
This shows that in the category of locally convex TVSs, Banach spaces are exactly those complete spaces that are both metrizable and have metrizable strong dual spaces.

Completions

Every normed space can be isometrically embedded onto a dense vector subspace of Banach space, where this Banach space is called a of the normed space. This Hausdorff completion is unique up to isometric isomorphism. More precisely, for every normed space $X,$ there exist a Banach space $Y$ and a mapping $T : X \to Y$ such that $T$ is an isometric mapping and $T\left(X\right)$ is dense in $Y.$ If $Z$ is another Banach space such that there is an isometric isomorphism from $X$ onto a dense subset of $Z,$ then $Z$ is isometrically isomorphic to $Y.$ This Banach space $Y$ is the Hausdorff of the normed space $X.$ The underlying metric space for $Y$ is the same as the metric completion of $X,$ with the vector space operations extended from $X$ to $Y.$ The completion of $X$ is sometimes denoted by $\widehat.$

General theory

Linear operators, isomorphisms

If $X$ and $Y$ are normed spaces over the same ground field $\mathbb,$ the set of all continuous $\mathbb$-linear maps $T : X \to Y$ is denoted by $B\left(X, Y\right).$ In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space $X$ to another normed space is continuous if and only if it is bounded on the closed unit ball of $X.$ Thus, the vector space $B\left(X, Y\right)$ can be given the operator norm $\, T\, = \sup \left\.$ For $Y$ a Banach space, the space $B\left(X, Y\right)$ is a Banach space with respect to this norm. In categorical contexts, it is sometimes convenient to restrict the
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
between two Banach spaces to only the short maps; in that case the space $B\left(X,Y\right)$ reappears as a natural
bifunctor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
. If $X$ is a Banach space, the space $B\left(X\right) = B\left(X, X\right)$ forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps. If $X$ and $Y$ are normed spaces, they are isomorphic normed spaces if there exists a linear bijection $T : X \to Y$ such that $T$ and its inverse $T^$ are continuous. If one of the two spaces $X$ or $Y$ is complete (or reflexive, separable, etc.) then so is the other space. Two normed spaces $X$ and $Y$ are isometrically isomorphic if in addition, $T$ is an isometry, that is, $\, T\left(x\right)\, = \, x\,$ for every $x$ in $X.$ The Banach–Mazur distance $d\left(X, Y\right)$ between two isomorphic but not isometric spaces $X$ and $Y$ gives a measure of how much the two spaces $X$ and $Y$ differ.

Continuous and bounded linear functions and seminorms

Every
continuous linear operator In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear ...
is a bounded linear operator and if dealing only with normed spaces then the converse is also true. That is, a linear operator between two normed spaces is bounded if and only if it is a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
. So in particular, because the scalar field (which is $\R$ or $\Complex$) is a normed space, a
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
on a normed space is a bounded linear functional if and only if it is a continuous linear functional. This allows for continuity-related results (like those below) to be applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used when dealing primarily with Banach spaces. If $f : X \to \R$ is a subadditive function (such as a norm, a
sublinear function In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a semin ...
, or real linear functional), then $f$ is continuous at the origin if and only if $f$ is uniformly continuous on all of $X$; and if in addition $f\left(0\right) = 0$ then $f$ is continuous if and only if its
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
$, f, : X \to \left[0, \infty\right)$ is continuous, which happens if and only if $\$ is an open subset of $X.$The fact that $\$ being open implies that $f : X \to \R$ is continuous simplifies proving continuity because this means that it suffices to show that $\$ is open for $r := 1$ and at $x_0 := 0$ (where $f\left(0\right) = 0$) rather than showing this for real $r > 0$ and $x_0 \in X.$ And very importantly for applying the Hahn–Banach theorem, a linear functional $f$ is continuous if and only if this is true of its real part $\operatorname f$ and moreover, $\, \operatorname f\, = \, f\,$ and Real and imaginary parts of a linear functional, the real part $\operatorname f$ completely determines $f,$ which is why the Hahn–Banach theorem is often stated only for real linear functionals. Also, a linear functional $f$ on $X$ is continuous if and only if the seminorm $, f,$ is continuous, which happens if and only if there exists a continuous seminorm $p : X \to \R$ such that $, f, \leq p$; this last statement involving the linear functional $f$ and seminorm $p$ is encountered in many versions of the Hahn–Banach theorem.

Basic notions

The Cartesian product $X \times Y$ of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used, such as $\, (x, y)\, _1 = \, x\, + \, y\, , \qquad \, (x, y)\, _\infty = \max (\, x\, , \, y\, )$ which correspond (respectively) to the
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The copr ...
and
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Prod ...
in the category of Banach spaces and short maps (discussed above). For finite (co)products, these norms give rise to isomorphic normed spaces, and the product $X \times Y$ (or the direct sum $X \oplus Y$) is complete if and only if the two factors are complete. If $M$ is a closed
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, ...
of a normed space $X,$ there is a natural norm on the quotient space $X / M,$ $\, x + M\, = \inf\limits_ \, x + m\, .$ The quotient $X / M$ is a Banach space when $X$ is complete.see pp. 17–19 in . The quotient map from $X$ onto $X / M,$ sending $x \in X$ to its class $x + M,$ is linear, onto and has norm $1,$ except when $M = X,$ in which case the quotient is the null space. The closed linear subspace $M$ of $X$ is said to be a complemented subspace of $X$ if $M$ is the range of a surjective bounded linear projection $P : X \to M.$ In this case, the space $X$ is isomorphic to the direct sum of $M$ and $\ker P,$ the kernel of the projection $P.$ Suppose that $X$ and $Y$ are Banach spaces and that $T \in B\left(X, Y\right).$ There exists a canonical factorization of $T$ as $T = T_1 \circ \pi, \ \ \ T : X \ \overset\ X / ker(T) \ \overset \ Y$ where the first map $\pi$ is the quotient map, and the second map $T_1$ sends every class $x + \ker T$ in the quotient to the image $T\left(x\right)$ in $Y.,$ This is well defined because all elements in the same class have the same image. The mapping $T_1$ is a linear bijection from $X / \ker T$ onto the range $T\left(X\right),$ whose inverse need not be bounded.

Classical spaces

Basic examples of Banach spaces include: the
Lp space In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbak ...
s $L^p$ and their special cases, the sequence spaces $\ell^p$ that consist of scalar sequences indexed by
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
s $\N$; among them, the space $\ell^1$ of absolutely summable sequences and the space $\ell^2$ of square summable sequences; the space $c_0$ of sequences tending to zero and the space $\ell^$ of bounded sequences; the space $C\left(K\right)$ of continuous scalar functions on a compact Hausdorff space $K,$ equipped with the max norm, $\, f\, _ = \max \, \quad f \in C(K).$ According to the Banach–Mazur theorem, every Banach space is isometrically isomorphic to a subspace of some $C\left(K\right).$ For every separable Banach space $X,$ there is a closed subspace $M$ of $\ell^1$ such that $X := \ell^1 / M.$ Any
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
serves as an example of a Banach space. A Hilbert space $H$ on $\mathbb = \Reals, \Complex$ is complete for a norm of the form $\, x\, _H = \sqrt,$ where $\langle \cdot, \cdot \rangle : H \times H \to \mathbb$ is the inner product, linear in its first argument that satisfies the following: $\begin \langle y, x \rangle &= \overline, \quad \text x, y \in H \\ \langle x, x \rangle & \geq 0, \quad \text x \in H \\ \langle x,x \rangle = 0 \text x &= 0. \end$ For example, the space $L^2$ is a Hilbert space. The Hardy spaces, the Sobolev spaces are examples of Banach spaces that are related to $L^p$ spaces and have additional structure. They are important in different branches of analysis, Harmonic analysis and
Partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
s among others.

Banach algebras

A Banach algebra is a Banach space $A$ over $\mathbb = \R$ or $\Complex,$ together with a structure of algebra over $\mathbb$, such that the product map $A \times A \ni \left(a, b\right) \mapsto ab \in A$ is continuous. An equivalent norm on $A$ can be found so that $\, ab\, \leq \, a\, \, b\,$ for all $a, b \in A.$

Examples

* The Banach space $C\left(K\right)$ with the pointwise product, is a Banach algebra. * The disk algebra $A\left(\mathbf\right)$ consists of functions
holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex der ...
in the open unit disk $D \subseteq \Complex$ and continuous on its closure: $\overline.$ Equipped with the max norm on $\overline,$ the disk algebra $A\left(\mathbf\right)$ is a closed subalgebra of $C\left\left(\overline\right\right).$ * The Wiener algebra $A\left(\mathbf\right)$ is the algebra of functions on the unit circle $\mathbf$ with absolutely convergent Fourier series. Via the map associating a function on $\mathbf$ to the sequence of its Fourier coefficients, this algebra is isomorphic to the Banach algebra $\ell^1\left(Z\right),$ where the product is the convolution of sequences. * For every Banach space $X,$ the space $B\left(X\right)$ of bounded linear operators on $X,$ with the composition of maps as product, is a Banach algebra. * A
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continu ...
is a complex Banach algebra $A$ with an antilinear involution $a \mapsto a^*$ such that $\left\, a^* a\right\, = \, a\, ^2.$ The space $B\left(H\right)$ of bounded linear operators on a Hilbert space $H$ is a fundamental example of C*-algebra. The Gelfand–Naimark theorem states that every C*-algebra is isometrically isomorphic to a C*-subalgebra of some $B\left(H\right).$ The space $C\left(K\right)$ of complex continuous functions on a compact Hausdorff space $K$ is an example of commutative C*-algebra, where the involution associates to every function $f$ its
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
$\overline.$

Dual space

If $X$ is a normed space and $\mathbb$ the underlying field (either the real or the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s), the continuous dual space is the space of continuous linear maps from $X$ into $\mathbb,$ or continuous linear functionals. The notation for the continuous dual is $X^ = B\left(X, \mathbb\right)$ in this article. Since $\mathbb$ is a Banach space (using the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
as norm), the dual $X^$ is a Banach space, for every normed space $X.$ The main tool for proving the existence of continuous linear functionals is the Hahn–Banach theorem. In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to the whole space, without increasing the norm of the functional. An important special case is the following: for every vector $x$ in a normed space $X,$ there exists a continuous linear functional $f$ on $X$ such that $f(x) = \, x\, _X, \quad \, f\, _ \leq 1.$ When $x$ is not equal to the $\mathbf$ vector, the functional $f$ must have norm one, and is called a norming functional for $x.$ The Hahn–Banach separation theorem states that two disjoint non-empty
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
s in a real Banach space, one of them open, can be separated by a closed affine
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hype ...
. The open convex set lies strictly on one side of the hyperplane, the second convex set lies on the other side but may touch the hyperplane. A subset $S$ in a Banach space $X$ is total if the
linear span In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
of $S$ is dense in $X.$ The subset $S$ is total in $X$ if and only if the only continuous linear functional that vanishes on $S$ is the $\mathbf$ functional: this equivalence follows from the Hahn–Banach theorem. If $X$ is the direct sum of two closed linear subspaces $M$ and $N,$ then the dual $X^$ of $X$ is isomorphic to the direct sum of the duals of $M$ and $N.$see p. 19 in . If $M$ is a closed linear subspace in $X,$ one can associate the $M$ in the dual, $M^ = \left\.$ The orthogonal $M^$ is a closed linear subspace of the dual. The dual of $M$ is isometrically isomorphic to $X\text{'} / M^.$ The dual of $X / M$ is isometrically isomorphic to $M^.$ The dual of a separable Banach space need not be separable, but: When $X\text{'}$ is separable, the above criterion for totality can be used for proving the existence of a countable total subset in $X.$

Weak topologies

The weak topology on a Banach space $X$ is the coarsest topology on $X$ for which all elements $x^$ in the continuous dual space $X^$ are continuous. The norm topology is therefore finer than the weak topology. It follows from the Hahn–Banach separation theorem that the weak topology is Hausdorff, and that a norm-closed convex subset of a Banach space is also weakly closed. A norm-continuous linear map between two Banach spaces $X$ and $Y$ is also weakly continuous, that is, continuous from the weak topology of $X$ to that of $Y.$ If $X$ is infinite-dimensional, there exist linear maps which are not continuous. The space $X^*$ of all linear maps from $X$ to the underlying field $\mathbb$ (this space $X^*$ is called the algebraic dual space, to distinguish it from $X^$ also induces a topology on $X$ which is finer than the weak topology, and much less used in functional analysis. On a dual space $X^,$ there is a topology weaker than the weak topology of $X^,$ called weak* topology. It is the coarsest topology on $X^$ for which all evaluation maps $x^ \in X^ \mapsto x^\left(x\right),$ where $x$ ranges over $X,$ are continuous. Its importance comes from the
Banach–Alaoglu theorem In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common p ...
. The Banach–Alaoglu theorem can be proved using Tychonoff's theorem about infinite products of compact Hausdorff spaces. When $X$ is separable, the unit ball $B^$ of the dual is a metrizable compact in the weak* topology.see Theorem 2.6.23, p. 231 in .

Examples of dual spaces

The dual of $c_0$ is isometrically isomorphic to $\ell^1$: for every bounded linear functional $f$ on $c_0,$ there is a unique element $y = \left\ \in \ell^1$ such that $f(x) = \sum_ x_n y_n, \qquad x = \ \in c_0, \ \ \text \ \ \, f\, _ = \, y\, _.$ The dual of $\ell^1$ is isometrically isomorphic to $\ell^$. The dual of Lebesgue space is isometrically isomorphic to when $1 \leq p < \infty$ and $\frac + \frac = 1.$ For every vector $y$ in a Hilbert space $H,$ the mapping $x \in H \to f_y(x) = \langle x, y \rangle$ defines a continuous linear functional $f_y$ on $H.$The Riesz representation theorem states that every continuous linear functional on $H$ is of the form $f_y$ for a uniquely defined vector $y$ in $H.$ The mapping $y \in H \to f_y$ is an antilinear isometric bijection from $H$ onto its dual $H\text{'}.$ When the scalars are real, this map is an isometric isomorphism. When $K$ is a compact Hausdorff topological space, the dual $M\left(K\right)$ of $C\left(K\right)$ is the space of
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
s in the sense of Bourbaki. The subset $P\left(K\right)$ of $M\left(K\right)$ consisting of non-negative measures of mass 1 ( probability measures) is a convex w*-closed subset of the unit ball of $M\left(K\right).$ The
extreme point In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex or ...
s of $P\left(K\right)$ are the Dirac measures on $K.$ The set of Dirac measures on $K,$ equipped with the w*-topology, is homeomorphic to $K.$ The result has been extended by Amir and Cambern to the case when the multiplicative Banach–Mazur distance between $C\left(K\right)$ and $C\left(L\right)$ is $< 2.$ The theorem is no longer true when the distance is $= 2.$ In the commutative Banach algebra $C\left(K\right),$ the maximal ideals are precisely kernels of Dirac measures on $K,$ $I_x = \ker \delta_x = \, \quad x \in K.$ More generally, by the Gelfand–Mazur theorem, the maximal ideals of a unital commutative Banach algebra can be identified with its characters—not merely as sets but as topological spaces: the former with the
hull-kernel topology In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra ''A'', denoted ''Â'', is the set of unitary representation, unitary equivalence classes of irreducible representation, irreducible *-representations of ''A''. A *-representation ...
and the latter with the w*-topology. In this identification, the maximal ideal space can be viewed as a w*-compact subset of the unit ball in the dual $A\text{'}.$ Not every unital commutative Banach algebra is of the form $C\left(K\right)$ for some compact Hausdorff space $K.$ However, this statement holds if one places $C\left(K\right)$ in the smaller category of commutative
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continu ...
s. Gelfand's representation theorem for commutative C*-algebras states that every commutative unital ''C''*-algebra $A$ is isometrically isomorphic to a $C\left(K\right)$ space. The Hausdorff compact space $K$ here is again the maximal ideal space, also called the
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of $A$ in the C*-algebra context.

Bidual

If $X$ is a normed space, the (continuous) dual $X\text{'}\text{'}$ of the dual $X\text{'}$ is called , or of $X.$ For every normed space $X,$ there is a natural map,
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
. For example, the dual of $X = c_0$ is identified with $\ell^1,$ and the dual of $\ell^1$ is identified with $\ell^,$ the space of bounded scalar sequences. Under these identifications, $F_X$ is the inclusion map from $c_0$ to $\ell^.$ It is indeed isometric, but not onto. If $F_X$ is surjective, then the normed space $X$ is called reflexive (see below). Being the dual of a normed space, the bidual $X\text{'}\text{'}$ is complete, therefore, every reflexive normed space is a Banach space. Using the isometric embedding $F_X,$ it is customary to consider a normed space $X$ as a subset of its bidual. When $X$ is a Banach space, it is viewed as a closed linear subspace of $X^.$ If $X$ is not reflexive, the unit ball of $X$ is a proper subset of the unit ball of $X^.$ The Goldstine theorem states that the unit ball of a normed space is weakly*-dense in the unit ball of the bidual. In other words, for every $x\text{'}\text{'}$ in the bidual, there exists a net $\left\left(x_i\right\right)_$ in $X$ so that

Banach's theorems

Here are the main general results about Banach spaces that go back to the time of Banach's book () and are related to the
Baire category theorem The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that t ...
. According to this theorem, a complete metric space (such as a Banach space, a Fréchet space or an F-space) cannot be equal to a union of countably many closed subsets with empty interiors. Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable
Hamel basis In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as component ...
is finite-dimensional. The Banach–Steinhaus theorem is not limited to Banach spaces. It can be extended for example to the case where $X$ is a Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood $U$ of $\mathbf$ in $X$ such that all $T$ in $F$ are uniformly bounded on $U,$ $\sup_ \sup_ \; \, T(x)\, _Y < \infty.$ This result is a direct consequence of the preceding ''Banach isomorphism theorem'' and of the canonical factorization of bounded linear maps. This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from $M_1 \oplus \cdots \oplus M_n$ onto $X$ sending $m_1, \cdots, m_n$ to the sum $m_1 + \cdots + m_n.$

Reflexivity

The normed space $X$ is called reflexive when the natural map $\begin F_X : X \to X'' \\ F_X(x) (f) = f(x) & \text x \in X, \text f \in X'\end$ is surjective. Reflexive normed spaces are Banach spaces. This is a consequence of the Hahn–Banach theorem. Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space $X$ onto the Banach space $Y,$ then $Y$ is reflexive. Indeed, if the dual $Y^$ of a Banach space $Y$ is separable, then $Y$ is separable. If $X$ is reflexive and separable, then the dual of $X^$ is separable, so $X^$ is separable. Hilbert spaces are reflexive. The $L^p$ spaces are reflexive when $1 < p < \infty.$ More generally, uniformly convex spaces are reflexive, by the Milman–Pettis theorem. The spaces are not reflexive. In these examples of non-reflexive spaces $X,$ the bidual $X\text{'}\text{'}$ is "much larger" than $X.$ Namely, under the natural isometric embedding of $X$ into $X\text{'}\text{'}$ given by the Hahn–Banach theorem, the quotient $X^ / X$ is infinite-dimensional, and even nonseparable. However, Robert C. James has constructed an example of a non-reflexive space, usually called "''the James space''" and denoted by $J,$ such that the quotient $J^ / J$ is one-dimensional. Furthermore, this space $J$ is isometrically isomorphic to its bidual. When $X$ is reflexive, it follows that all closed and bounded convex subsets of $X$ are weakly compact. In a Hilbert space $H,$ the weak compactness of the unit ball is very often used in the following way: every bounded sequence in $H$ has weakly convergent subsequences. Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain optimization problems. For example, every
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
continuous function on the unit ball $B$ of a reflexive space attains its minimum at some point in $B.$ As a special case of the preceding result, when $X$ is a reflexive space over $\R,$ every continuous linear functional $f$ in $X^$ attains its maximum $\, f\,$ on the unit ball of $X.$ The following theorem of Robert C. James provides a converse statement. The theorem can be extended to give a characterization of weakly compact convex sets. On every non-reflexive Banach space $X,$ there exist continuous linear functionals that are not ''norm-attaining''. However, the
Bishop A bishop is an ordained clergy member who is entrusted with a position of authority and oversight in a religious institution. In Christianity, bishops are normally responsible for the governance of dioceses. The role or office of bishop is ...
Phelps theorem states that norm-attaining functionals are norm dense in the dual $X^$ of $X.$

Weak convergences of sequences

A sequence $\left\$ in a Banach space $X$ is weakly convergent to a vector $x \in X$ if $\left\$ converges to $f\left(x\right)$ for every continuous linear functional $f$ in the dual $X^.$ The sequence $\left\$ is a weakly Cauchy sequence if $\left\$ converges to a scalar limit $L\left(f\right),,$ for every $f$ in $X^.$ A sequence $\left\$ in the dual $X^$ is weakly* convergent to a functional $f \in X^$ if $f_n\left(x\right)$ converges to $f\left(x\right)$ for every $x$ in $X.$ Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the Banach–Steinhaus theorem. When the sequence $\left\$ in $X$ is a weakly Cauchy sequence, the limit $L$ above defines a bounded linear functional on the dual $X^,$ that is, an element $L$ of the bidual of $X,$ and $L$ is the limit of $\left\$ in the weak*-topology of the bidual. The Banach space $X$ is weakly sequentially complete if every weakly Cauchy sequence is weakly convergent in $X.$ It follows from the preceding discussion that reflexive spaces are weakly sequentially complete. An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equal to the $\mathbf$ vector. The unit vector basis of $\ell^p$ for $1 < p < \infty,$ or of $c_0,$ is another example of a weakly null sequence, that is, a sequence that converges weakly to $\mathbf.$ For every weakly null sequence in a Banach space, there exists a sequence of convex combinations of vectors from the given sequence that is norm-converging to $\mathbf.$ The unit vector basis of $\ell^1$ is not weakly Cauchy. Weakly Cauchy sequences in $\ell^1$ are weakly convergent, since $L^1$-spaces are weakly sequentially complete. Actually, weakly convergent sequences in $\ell^1$ are norm convergent. This means that $\ell^1$ satisfies Schur's property.

Results involving the $\ell^1$ basis

Weakly Cauchy sequences and the $\ell^1$ basis are the opposite cases of the dichotomy established in the following deep result of H. P. Rosenthal. A complement to this result is due to Odell and Rosenthal (1975). By the Goldstine theorem, every element of the unit ball $B^$ of $X^$ is weak*-limit of a net in the unit ball of $X.$ When $X$ does not contain $\ell^1,$ every element of $B^$ is weak*-limit of a in the unit ball of $X.$ When the Banach space $X$ is separable, the unit ball of the dual $X^,$ equipped with the weak*-topology, is a metrizable compact space $K,$ and every element $x^$ in the bidual $X^$ defines a bounded function on $K$: $x' \in K \mapsto x''(x'), \quad \left , x''(x')\ \leq \left \, x''\right \, .$ This function is continuous for the compact topology of $K$ if and only if $x^$ is actually in $X,$ considered as subset of $X^.$ Assume in addition for the rest of the paragraph that $X$ does not contain $\ell^1.$ By the preceding result of Odell and Rosenthal, the function $x^$ is the pointwise limit on $K$ of a sequence $\left\ \subseteq X$ of continuous functions on $K,$ it is therefore a first Baire class function on $K.$ The unit ball of the bidual is a pointwise compact subset of the first Baire class on $K.$

Sequences, weak and weak* compactness

When $X$ is separable, the unit ball of the dual is weak*-compact by the
Banach–Alaoglu theorem In functional analysis and related branches of mathematics, the Banach–Alaoglu theorem (also known as Alaoglu's theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common p ...
and metrizable for the weak* topology, hence every bounded sequence in the dual has weakly* convergent subsequences. This applies to separable reflexive spaces, but more is true in this case, as stated below. The weak topology of a Banach space $X$ is metrizable if and only if $X$ is finite-dimensional. If the dual $X\text{'}$ is separable, the weak topology of the unit ball of $X$ is metrizable. This applies in particular to separable reflexive Banach spaces. Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences. A Banach space $X$ is reflexive if and only if each bounded sequence in $X$ has a weakly convergent subsequence. A weakly compact subset $A$ in $\ell^1$ is norm-compact. Indeed, every sequence in $A$ has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of $\ell^1.$

Schauder bases

A Schauder basis in a Banach space $X$ is a sequence $\left\_$ of vectors in $X$ with the property that for every vector $x \in X,$ there exist defined scalars $\left\_$ depending on $x,$ such that $x = \sum_^ x_n e_n, \quad \textit \quad x = \lim_n P_n(x), \ P_n(x) := \sum_^n x_k e_k.$ Banach spaces with a Schauder basis are necessarily separable, because the countable set of finite linear combinations with rational coefficients (say) is dense. It follows from the Banach–Steinhaus theorem that the linear mappings $\left\$ are uniformly bounded by some constant $C.$ Let $\left\$ denote the coordinate functionals which assign to every $x$ in $X$ the coordinate $x_n$ of $x$ in the above expansion. They are called biorthogonal functionals. When the basis vectors have norm $1,$ the coordinate functionals $\left\$ have norm $\,\leq 2 C$ in the dual of $X.$ Most classical separable spaces have explicit bases. The Haar system $\left\$ is a basis for The trigonometric system is a basis in $L^p\left(\mathbf\right)$ when $1 < p < \infty.$ The Schauder system is a basis in the space The question of whether the disk algebra $A\left(\mathbf\right)$ has a basis remained open for more than forty years, until Bočkarev showed in 1974 that $A\left(\mathbf\right)$ admits a basis constructed from the Franklin system. Since every vector $x$ in a Banach space $X$ with a basis is the limit of $P_n\left(x\right),$ with $P_n$ of finite rank and uniformly bounded, the space $X$ satisfies the bounded approximation property. The first example by Enflo of a space failing the approximation property was at the same time the first example of a separable Banach space without a Schauder basis. Robert C. James characterized reflexivity in Banach spaces with a basis: the space $X$ with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete. In this case, the biorthogonal functionals form a basis of the dual of $X.$

Tensor product

Let $X$ and $Y$ be two $\mathbb$-vector spaces. The
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes ...
$X \otimes Y$ of $X$ and $Y$ is a $\mathbb$-vector space $Z$ with a bilinear mapping $T : X \times Y \to Z$ which has the following universal property: :If $T_1 : X \times Y \to Z_1$ is any bilinear mapping into a $\mathbb$-vector space $Z_1,$ then there exists a unique linear mapping $f : Z \to Z_1$ such that $T_1 = f \circ T.$ The image under $T$ of a couple $\left(x, y\right)$ in $X \times Y$ is denoted by $x \otimes y,$ and called a simple tensor. Every element $z$ in $X \otimes Y$ is a finite sum of such simple tensors. There are various norms that can be placed on the tensor product of the underlying vector spaces, amongst others the projective cross norm and injective cross norm introduced by A. Grothendieck in 1955. In general, the tensor product of complete spaces is not complete again. When working with Banach spaces, it is customary to say that the projective tensor product of two Banach spaces $X$ and $Y$ is the $X \widehat_\pi Y$ of the algebraic tensor product $X \otimes Y$ equipped with the projective tensor norm, and similarly for the injective tensor product $X \widehat_\varepsilon Y.$ Grothendieck proved in particular that where $K$ is a compact Hausdorff space, $C\left(K, Y\right)$ the Banach space of continuous functions from $K$ to $Y$ and the space of Bochner-measurable and integrable functions from

Tensor products and the approximation property

Let $X$ be a Banach space. The tensor product $X\text{'} \widehat \otimes_\varepsilon X$ is identified isometrically with the closure in $B\left(X\right)$ of the set of finite rank operators. When $X$ has the approximation property, this closure coincides with the space of
compact operator In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...
s on $X.$ For every Banach space $Y,$ there is a natural norm $1$ linear map $Y \widehat\otimes_\pi X \to Y \widehat\otimes_\varepsilon X$ obtained by extending the identity map of the algebraic tensor product. Grothendieck related the approximation problem to the question of whether this map is one-to-one when $Y$ is the dual of $X.$ Precisely, for every Banach space $X,$ the map $X' \widehat \otimes_\pi X \ \longrightarrow X' \widehat \otimes_\varepsilon X$ is one-to-one if and only if $X$ has the approximation property. Grothendieck conjectured that $X \widehat_\pi Y$ and $X \widehat_\varepsilon Y$ must be different whenever $X$ and $Y$ are infinite-dimensional Banach spaces. This was disproved by Gilles Pisier in 1983. Pisier constructed an infinite-dimensional Banach space $X$ such that $X \widehat_\pi X$ and $X \widehat_\varepsilon X$ are equal. Furthermore, just as Enflo's example, this space $X$ is a "hand-made" space that fails to have the approximation property. On the other hand, Szankowski proved that the classical space $B\left\left(\ell^2\right\right)$ does not have the approximation property.

Some classification results

Characterizations of Hilbert space among Banach spaces

A necessary and sufficient condition for the norm of a Banach space $X$ to be associated to an inner product is the parallelogram identity: It follows, for example, that the Lebesgue space is a Hilbert space only when $p = 2.$ If this identity is satisfied, the associated inner product is given by the polarization identity. In the case of real scalars, this gives: $\langle x, y\rangle = \tfrac \left(\, x+y\, ^2 - \, x-y\, ^2 \right).$ For complex scalars, defining the inner product so as to be $\Complex$-linear in $x,$ antilinear in $y,$ the polarization identity gives: $\langle x,y\rangle = \tfrac \left(\, x+y\, ^2 - \, x-y\, ^2 + i \left(\, x+iy\, ^2 - \, x-iy\, ^2\right)\right).$ To see that the parallelogram law is sufficient, one observes in the real case that $\langle x, y \rangle$ is symmetric, and in the complex case, that it satisfies the Hermitian symmetry property and $\langle i x, y \rangle = i \langle x, y \rangle.$ The parallelogram law implies that $\langle x, y \rangle$ is additive in $x.$ It follows that it is linear over the rationals, thus linear by continuity. Several characterizations of spaces isomorphic (rather than isometric) to Hilbert spaces are available. The parallelogram law can be extended to more than two vectors, and weakened by the introduction of a two-sided inequality with a constant $c \geq 1$: Kwapień proved that if $c^ \sum_^n \left\, x_k\right\, ^2 \leq \operatorname_ \left\, \sum_^n \pm x_k\right\, ^2 \leq c^2 \sum_^n \left\, x_k\right\, ^2$ for every integer $n$ and all families of vectors$\left\ \subseteq X,$ then the Banach space $X$ is isomorphic to a Hilbert space. Here, $\operatorname_$ denotes the average over the $2^n$ possible choices of signs $\pm 1.$ In the same article, Kwapień proved that the validity of a Banach-valued Parseval's theorem for the Fourier transform characterizes Banach spaces isomorphic to Hilbert spaces. Lindenstrauss and Tzafriri proved that a Banach space in which every closed linear subspace is complemented (that is, is the range of a bounded linear projection) is isomorphic to a Hilbert space. The proof rests upon Dvoretzky's theorem about Euclidean sections of high-dimensional centrally symmetric convex bodies. In other words, Dvoretzky's theorem states that for every integer $n,$ any finite-dimensional normed space, with dimension sufficiently large compared to $n,$ contains subspaces nearly isometric to the $n$-dimensional Euclidean space. The next result gives the solution of the so-called . An infinite-dimensional Banach space $X$ is said to be homogeneous if it is isomorphic to all its infinite-dimensional closed subspaces. A Banach space isomorphic to $\ell^2$ is homogeneous, and Banach asked for the converse. An infinite-dimensional Banach space is hereditarily indecomposable when no subspace of it can be isomorphic to the direct sum of two infinite-dimensional Banach spaces. The Gowers dichotomy theorem asserts that every infinite-dimensional Banach space $X$ contains, either a subspace $Y$ with unconditional basis, or a hereditarily indecomposable subspace $Z,$ and in particular, $Z$ is not isomorphic to its closed hyperplanes. If $X$ is homogeneous, it must therefore have an unconditional basis. It follows then from the partial solution obtained by Komorowski and Tomczak–Jaegermann, for spaces with an unconditional basis, that $X$ is isomorphic to $\ell^2.$

Metric classification

If $T : X \to Y$ is an isometry from the Banach space $X$ onto the Banach space $Y$ (where both $X$ and $Y$ are vector spaces over $\R$), then the Mazur–Ulam theorem states that $T$ must be an affine transformation. In particular, if $T\left(0_X\right) = 0_Y,$ this is $T$ maps the zero of $X$ to the zero of $Y,$ then $T$ must be linear. This result implies that the metric in Banach spaces, and more generally in normed spaces, completely captures their linear structure.

Topological classification

Finite dimensional Banach spaces are homeomorphic as topological spaces, if and only if they have the same dimension as real vector spaces. Anderson–Kadec theorem (1965–66) proves that any two infinite-dimensional separable Banach spaces are homeomorphic as topological spaces. Kadec's theorem was extended by Torunczyk, who proved that any two Banach spaces are homeomorphic if and only if they have the same density character, the minimum cardinality of a dense subset.

Spaces of continuous functions

When two compact Hausdorff spaces $K_1$ and $K_2$ are homeomorphic, the Banach spaces $C\left\left(K_1\right\right)$ and $C\left\left(K_2\right\right)$ are isometric. Conversely, when $K_1$ is not homeomorphic to $K_2,$ the (multiplicative) Banach–Mazur distance between $C\left\left(K_1\right\right)$ and $C\left\left(K_2\right\right)$ must be greater than or equal to $2,$ see above the results by Amir and Cambern. Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin: The situation is different for
countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
compact Hausdorff spaces. Every countably infinite compact $K$ is homeomorphic to some closed interval of
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least n ...
s $\langle 1, \alpha \rangle = \$ equipped with the
order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, t ...
, where $\alpha$ is a countably infinite ordinal. The Banach space $C\left(K\right)$ is then isometric to . When $\alpha, \beta$ are two countably infinite ordinals, and assuming $\alpha \leq \beta,$ the spaces and are isomorphic if and only if .Bessaga, Czesław; Pełczyński, Aleksander (1960), "Spaces of continuous functions. IV. On isomorphical classification of spaces of continuous functions", Studia Math. 19:53–62. For example, the Banach spaces $C(\langle 1, \omega\rangle), \ C(\langle 1, \omega^ \rangle), \ C(\langle 1, \omega^\rangle), \ C(\langle 1, \omega^ \rangle), \cdots, C(\langle 1, \omega^ \rangle), \cdots$ are mutually non-isomorphic.

Derivatives

Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gateaux derivative for details. The Fréchet derivative allows for an extension of the concept of a
total derivative In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with res ...
to Banach spaces. The Gateaux derivative allows for an extension of a
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity s ...
to locally convex
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s. Fréchet differentiability is a stronger condition than Gateaux differentiability. The quasi-derivative is another generalization of directional derivative that implies a stronger condition than Gateaux differentiability, but a weaker condition than Fréchet differentiability.

Generalizations

Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions $\R \to \R,$ or the space of all distributions on $\R,$ are complete but are not normed vector spaces and hence not Banach spaces. In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces.

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Bibliography

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