In . Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but converse is not true. For example, the set of the

topological vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

then the following are equivalent:
# $X$ is normable.
# $X$ has a bounded neighborhood of the origin.
# the strong dual space $X^\_b$ of $X$ is normable.
# the strong dual space $X^\_b$ of $X$ is Metrizable topological vector space, metrizable.
Furthermore, $X$ is finite dimensional if and only if $X^\_$ is normable (here $X^\_$ denotes $X^$ endowed with the weak-* topology).
The topology $\backslash tau$ of the Fréchet space $C^(K),$ as defined in the article on spaces of test functions and distributions, is defined by a countable family of norms but it is a normable space because there does not exist any norm $\backslash ,\; \backslash cdot\backslash ,$ on $C^(K)$ such that the topology that this norm induces is equal to $\backslash tau.$
Even if a metrizable topological vector space has a topology that is defined by a family of norms, then it may nevertheless still fail to be normable space (meaning that its topology can not be defined by any norm).
An example of such a space is the Fréchet space $C^(K),$ whose definition can be found in the article on spaces of test functions and distributions, because its topology $\backslash tau$ is defined by a countable family of norms but it is a normable space because there does not exist any norm $\backslash ,\; \backslash cdot\backslash ,$ on $C^(K)$ such that the topology this norm induces is equal to $\backslash tau.$
In fact, the topology of a Locally convex topological vector space, locally convex space $X$ can be a defined by a family of on $X$ if and only if there exists continuous norm on $X.$

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, a normed vector space or normed space is a vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

over the real
Real may refer to:
* Reality
Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...

or complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

numbers, on which a norm
Norm, the Norm or NORM may refer to:
In academic disciplines
* Norm (geology), an estimate of the idealised mineral content of a rock
* Norm (philosophy)
Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...

is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real world. A norm is a real-valued function
Mass measured in grams is a function from this collection of weight to positive number">positive
Positive is a property of Positivity (disambiguation), positivity and may refer to:
Mathematics and science
* Converging lens or positive lens, i ...

defined on the vector space that is commonly denoted $x\backslash mapsto\; \backslash ,\; x\backslash ,\; ,$ and has the following properties:
#It is nonnegative, meaning that $\backslash ,\; x\backslash ,\; \backslash geq\; 0$ for every vector $x.$
#It is positive on nonzero vectors, that is, $$\backslash ,\; x\backslash ,\; =\; 0\; \backslash text\; x\; =\; 0.$$
# For every vector $x,$ and every scalar $\backslash alpha,$ $$\backslash ,\; \backslash alpha\; x\backslash ,\; =\; ,\; \backslash alpha,\; \backslash ,\; \backslash ,\; x\backslash ,\; .$$
# The holds; that is, for every vectors $x$ and $y,$ $$\backslash ,\; x+y\backslash ,\; \backslash leq\; \backslash ,\; x\backslash ,\; +\; \backslash ,\; y\backslash ,\; .$$
A norm induces a distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

, called its , by the formula
$$d(x,y)\; =\; \backslash ,\; y-x\backslash ,\; .$$
which make any normed vector space into a metric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

and a topological vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. If this metric $d$ is complete then the normed space is a Banach space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

finite sequence
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s of real numbers can be normed with the Euclidean norm
Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...

, but it is not complete for this norm.
An inner product space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

is a normed vector space whose norm is the square root of the inner product of a vector and itself. The Euclidean norm
Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...

of a Euclidean vector space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space
Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called p ...

is a special case that allows defining Euclidean distance by the formula
$$d(A,\; B)\; =\; \backslash ,\; \backslash overrightarrow\backslash ,\; .$$
The study of normed spaces and Banach spaces is a fundamental part of functional analysis, which is a major subfield of mathematics.
Definition

A normed vector space is avector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

equipped with a norm
Norm, the Norm or NORM may refer to:
In academic disciplines
* Norm (geology), an estimate of the idealised mineral content of a rock
* Norm (philosophy)
Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...

. A is a vector space equipped with a seminorm.
A useful Triangle inequality#Reverse triangle inequality, variation of the triangle inequality is
$$\backslash ,\; x-y\backslash ,\; \backslash geq\; ,\; \backslash ,\; x\backslash ,\; -\; \backslash ,\; y\backslash ,\; ,$$
for any vectors $x$ and $y.$
This also shows that a vector norm is a continuous function.
Property 3 depends on a choice of norm $,\; \backslash alpha,$ on the field of scalars. When the scalar field is $\backslash R$ (or more generally a subset of $\backslash Complex$), this is usually taken to be the ordinary absolute value, but other choices are possible. For example, for a vector space over $\backslash Q$ one could take $,\; \backslash alpha,$ to be the p-adic norm, $p$-adic norm.
Topological structure

If $(V,\; \backslash ,\; \backslash ,\backslash cdot\backslash ,\backslash ,\; )$ is a normed vector space, the norm $\backslash ,\; \backslash ,\backslash cdot\backslash ,\backslash ,$ induces a Metric (mathematics), metric (a notion of ''distance'') and therefore a topology on $V.$ This metric is defined in the natural way: the distance between two vectors $\backslash mathbf$ and $\backslash mathbf$ is given by $\backslash ,\; \backslash mathbf\; -\; \backslash mathbf\backslash ,\; .$ This topology is precisely the weakest topology which makes $\backslash ,\; \backslash ,\backslash cdot\backslash ,\backslash ,$ continuous and which is compatible with the linear structure of $V$ in the following sense: #The vector addition $\backslash ,+\backslash ,\; :\; V\; \backslash times\; V\; \backslash to\; V$ is jointly continuous with respect to this topology. This follows directly from the . #The scalar multiplication $\backslash ,\backslash cdot\backslash ,\; :\; \backslash mathbb\; \backslash times\; V\; \backslash to\; V,$ where $\backslash mathbb$ is the underlying scalar field of $V,$ is jointly continuous. This follows from the triangle inequality and homogeneity of the norm. Similarly, for any seminormed vector space we can define the distance between two vectors $\backslash mathbf$ and $\backslash mathbf$ as $\backslash ,\; \backslash mathbf\; -\; \backslash mathbf\backslash ,\; .$ This turns the seminormed space into a pseudometric space (notice this is weaker than a metric) and allows the definition of notions such as Continuous function (topology), continuity and Limit of a function, convergence. To put it more abstractly every seminormed vector space is atopological vector space
In mathematics
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and thus carries a topological structure which is induced by the semi-norm.
Of special interest are Complete space, complete normed spaces, which are known as .
Every normed vector space $V$ sits as a dense subspace inside some Banach space; this Banach space is essentially uniquely defined by $V$ and is called the of $V.$
Two norms on the same vector space are called if they define the same Topology (structure), topology. On a finite-dimensional vector space, all norms are equivalent but this is not true for infinite dimensional vector spaces.
All norms on a finite-dimensional vector space are equivalent from a topological viewpoint as they induce the same topology (although the resulting metric spaces need not be the same)., Theorem 1.3.6 And since any Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are Banach spaces. A normed vector space $V$ is locally compact if and only if the unit ball $B\; =\; \backslash $ is Compact space, compact, which is the case if and only if $V$ is finite-dimensional; this is a consequence of Riesz's lemma. (In fact, a more general result is true: a topological vector space is locally compact if and only if it is finite-dimensional. The point here is that we don't assume the topology comes from a norm.)
The topology of a seminormed vector space has many nice properties. Given a neighbourhood system $\backslash mathcal(0)$ around 0 we can construct all other neighbourhood systems as
$$\backslash mathcal(x)\; =\; x\; +\; \backslash mathcal(0)\; :=\; \backslash $$
with
$$x\; +\; N\; :=\; \backslash .$$
Moreover, there exists a neighbourhood basis for the origin consisting of Absorbing set, absorbing and convex sets. As this property is very useful in functional analysis, generalizations of normed vector spaces with this property are studied under the name locally convex spaces.
A norm (or seminorm) $\backslash ,\; \backslash cdot\backslash ,$ on a topological vector space $(X,\; \backslash tau)$ is continuous if and only if the topology $\backslash tau\_$ that $\backslash ,\; \backslash cdot\backslash ,$ induces on $X$ is Comparison of topologies, coarser than $\backslash tau$ (meaning, $\backslash tau\_\; \backslash subseteq\; \backslash tau$), which happens if and only if there exists some open ball $B$ in $(X,\; \backslash ,\; \backslash cdot\backslash ,\; )$ (such as maybe $\backslash $ for example) that is open in $(X,\; \backslash tau)$ (said different, such that $B\; \backslash in\; \backslash tau$).
Normable spaces

Atopological vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

$(X,\; \backslash tau)$ is called normable if there exists a norm $\backslash ,\; \backslash cdot\; \backslash ,$ on $X$ such that the canonical metric $(x,\; y)\; \backslash mapsto\; \backslash ,\; y-x\backslash ,$ induces the topology $\backslash tau$ on $X.$
The following theorem is due to Andrey Kolmogorov, Kolmogorov:
Kolmogorov's normability criterion: A Hausdorff topological vector space is normable if and only if there exists a convex, von Neumann bounded neighborhood of $0\; \backslash in\; X.$
A product of a family of normable spaces is normable if and only if only finitely many of the spaces are non-trivial (that is, $\backslash neq\; \backslash $). Furthermore, the quotient of a normable space $X$ by a closed vector subspace $C$ is normable, and if in addition $X$'s topology is given by a norm $\backslash ,\; \backslash ,\backslash cdot,\backslash ,$ then the map $X/C\; \backslash to\; \backslash R$ given by $x\; +\; C\; \backslash mapsto\; \backslash inf\_\; \backslash ,\; x\; +\; c\backslash ,$ is a well defined norm on $X\; /\; C$ that induces the quotient topology on $X\; /\; C.$
If $X$ is a Hausdorff Locally convex topological vector space, locally convex Linear maps and dual spaces

The most important maps between two normed vector spaces are the Continuous function (topology), continuous Linear transformation, linear maps. Together with these maps, normed vector spaces form a Category theory, category. The norm is a continuous function on its vector space. All linear maps between finite dimensional vector spaces are also continuous. An ''isometry'' between two normed vector spaces is a linear map $f$ which preserves the norm (meaning $\backslash ,\; f(\backslash mathbf)\backslash ,\; =\; \backslash ,\; \backslash mathbf\backslash ,$ for all vectors $\backslash mathbf$). Isometries are always continuous and injective. A surjective isometry between the normed vector spaces $V$ and $W$ is called an ''isometric isomorphism'', and $V$ and $W$ are called ''isometrically isomorphic''. Isometrically isomorphic normed vector spaces are identical for all practical purposes. When speaking of normed vector spaces, we augment the notion of dual space to take the norm into account. The dual $V^$ of a normed vector space $V$ is the space of all ''continuous'' linear maps from $V$ to the base field (the complexes or the reals) — such linear maps are called "functionals". The norm of a functional $\backslash varphi$ is defined as the supremum of $,\; \backslash varphi(\backslash mathbf),$ where $\backslash mathbf$ ranges over all unit vectors (that is, vectors of norm $1$) in $V.$ This turns $V^$ into a normed vector space. An important theorem about continuous linear functionals on normed vector spaces is the Hahn–Banach theorem.Normed spaces as quotient spaces of seminormed spaces

The definition of many normed spaces (in particular,Banach space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s) involves a seminorm defined on a vector space and then the normed space is defined as the Quotient space (linear algebra), quotient space by the subspace of elements of seminorm zero. For instance, with the Lp space, $L^p$ spaces, the function defined by
$$\backslash ,\; f\backslash ,\; \_p\; =\; \backslash left(\; \backslash int\; ,\; f(x),\; ^p\; \backslash ;dx\; \backslash right)^$$
is a seminorm on the vector space of all functions on which the Lebesgue integral on the right hand side is defined and finite. However, the seminorm is equal to zero for any function Support (mathematics), supported on a set of Lebesgue measure zero. These functions form a subspace which we "quotient out", making them equivalent to the zero function.
Finite product spaces

Given $n$ seminormed spaces $\backslash left(X\_i,\; q\_i\backslash right)$ with seminorms $q\_i\; :\; X\_i\; \backslash to\; \backslash R,$ denote the product space by $$X\; :=\; \backslash prod\_^n\; X\_i$$ where vector addition defined as $$\backslash left(x\_1,\backslash ldots,x\_n\backslash right)\; +\; \backslash left(y\_1,\backslash ldots,y\_n\backslash right)\; :=\; \backslash left(x\_1\; +\; y\_1,\; \backslash ldots,\; x\_n\; +\; y\_n\backslash right)$$ and scalar multiplication defined as $$\backslash alpha\; \backslash left(x\_1,\backslash ldots,x\_n\backslash right)\; :=\; \backslash left(\backslash alpha\; x\_1,\; \backslash ldots,\; \backslash alpha\; x\_n\backslash right).$$ Define a new function $q\; :\; X\; \backslash to\; \backslash R$ by $$q\backslash left(x\_1,\backslash ldots,x\_n\backslash right)\; :=\; \backslash sum\_^n\; q\_i\backslash left(x\_i\backslash right),$$ which is a seminorm on $X.$ The function $q$ is a norm if and only if all $q\_i$ are norms. More generally, for each real $p\; \backslash geq\; 1$ the map $q\; :\; X\; \backslash to\; \backslash R$ defined by $$q\backslash left(x\_1,\backslash ldots,x\_n\backslash right)\; :=\; \backslash left(\backslash sum\_^n\; q\_i\backslash left(x\_i\backslash right)^p\backslash right)^$$ is a semi norm. For each $p$ this defines the same topological space. A straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising as the product space of a normed space and a space with trivial seminorm. Consequently, many of the more interesting examples and applications of seminormed spaces occur for infinite-dimensional vector spaces.See also

*Banach space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, normed vector spaces which are complete with respect to the metric induced by the norm
*
* Finsler manifold, where the length of each tangent vector is determined by a norm
* Inner product space, normed vector spaces where the norm is given by an inner product
*
* Locally convex topological vector space – a vector space with a topology defined by convex open sets
* Space (mathematics) – mathematical set with some added structure
*
References

Bibliography

* * * *External links

* {{DEFAULTSORT:Normed Vector Space Banach spaces, Normed spaces,