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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 on ...
, two methods of constructing
normed spaces 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" ...
from disks were systematically employed by Alexander Grothendieck to define
nuclear operator In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector space ...
s and
nuclear space In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, ...
s. One method is used if the disk D is bounded: in this case, the auxiliary normed space is \operatorname D with norm p_D(x) := \inf_ r. The other method is used if the disk D is absorbing: in this case, the auxiliary normed space is the quotient space X / p_D^(0). If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic (as
topological vector spaces 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 ...
and as
normed 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).


Preliminaries

A subset of a vector space is called a disk and is said to be disked, absolutely convex, or convex balanced if it is
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 polyto ...
and
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 ...
. If C and D are subsets of a vector space X then D absorbs C if there exists a real r > 0 such that C \subseteq a Dfor any scalar a satisfying , a, \geq r. WThe set D is called absorbing in X if D absorbs \ for every x \in X. A subset B of 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 al ...
(TVS) X is said to be bounded in X if every neighborhood of the origin in X absorbs B. A subset of a TVS X is called bornivorous if it absorbs all bounded subsets of X.


Induced by a bounded disk – Banach disks

Henceforth, X will be a real or complex vector space (not necessarily a TVS, yet) and D will be a disk in X.


Seminormed space induced by a disk

Let X will be a real or complex vector space. For any subset D of X, the
Minkowski functional In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, the ...
of D defined by: *If D = \varnothing then define p_(x) : \ \to absorbing in \operatorname D then denote the
Minkowski functional In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, the ...
of D in \operatorname D by p_D : \operatorname D \to [0, \infty) where for all x \in \operatorname D, this is defined by p_D (x) := \inf_ \left\. Let X will be a real or complex vector space. For any subset D of X such that the Minkowski functional p_Dis a
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
on \operatorname D, let X_D denote X_D := \left(\operatorname D, p_D\right) which is called the seminormed space induced by D where it is say "normed" if p_Dis 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 envi ...
. Assumption (Topology): X_D = \operatorname D is endowed with the seminorm topology induced by p_D, which will be denoted by \tau_D or \tau_ Importantly, this topology stems ''entirely'' from the set D, the algebraic structure of X, and the usual topology on \R (since p_Dis defined using the set D and scalar multiplication). This justifies the study of Banach disks and is part of the reason why they play an important role in the theory of
nuclear operator In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector space ...
s and
nuclear space In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, ...
s. The inclusion map \operatorname_D : X_D \to X is called the canonical map. Suppose that D is a disk. Then \operatorname D = \bigcup_^ n D so that D is absorbing in \operatorname D, 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 characteriz ...
of D. The set \ of all positive scalar multiples of D forms a basis of neighborhoods at 0 for 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 ve ...
topology \tau_D on \operatorname D. The
Minkowski functional In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, the ...
of the disk D in \operatorname D guarantees that p_Dis well-defined and forms a
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
on \operatorname D. The locally convex topology induced by this seminorm is the topology \tau_D that was defined before.


Banach disk definition

A bounded disk D in 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 al ...
X such that \left(X_D, p_D\right) is a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vect ...
is called a Banach disk, infracomplete, or a bounded completant in X. If its shown that \left(\operatorname D, p_D\right) is a Banach space then D will be a Banach disk in ''any'' TVS that contains D as a bounded subset. This is because the Minkowski functional p_Dis defined in purely algebraic terms. Consequently, the question of whether or not \left(X_D, p_D\right) forms a Banach space is dependent only on the disk D and the Minkowski functional p_D, and not on any particular TVS topology that X may carry. Thus the requirement that a Banach disk in a TVS X be a bounded subset of X is the only property that ties a Banach disk's topology to the topology of its containing TVS X.


Properties of disk induced seminormed spaces

;Bounded disks The following result explains why Banach disks are required to be bounded. ;Hausdorffness The space \left(X_D, p_D\right) is Hausdorff if and only if p_Dis a norm, which happens if and only if D does not contain any non-trivial vector subspace. In particular, if there exists a Hausdorff TVS topology on X such that D is bounded in X then p_Dis a norm. An example where X_D is not Hausdorff is obtained by letting X = \R^2 and letting D be the x-axis. ;Convergence of nets Suppose that D is a disk in X such that X_D is Hausdorff and let x_ = \left(x_i\right)_ be a net in X_D. Then x_ \to 0 in X_D if and only if there exists a net r_ = \left(r_i\right)_ of real numbers such that r_ \to 0 and x_i \in r_i D for all i; moreover, in this case it will be assumed without loss of generality that r_i \geq 0 for all i. ;Relationship between disk-induced spaces If C \subseteq D \subseteq Xthen \operatorname C \subseteq \operatorname D and p_D \leq p_C on \operatorname C, so define the following continuous linear map: If C and D are disks in X with C \subseteq D then call the inclusion map \operatorname_C^D : X_C \to X_D the canonical inclusion of X_C into X_D. In particular, the subspace topology that \operatorname C inherits from \left(X_D, p_D\right) is weaker than \left(X_C, p_C\right)'s seminorm topology. ;D as the closed unit ball The disk D is a closed subset of \left(X_D, p_D\right) if and only if D is the closed unit ball of the seminorm p_D; that is, D = \left\. If D is a disk in a vector space X and if there exists a TVS topology \tau on \operatorname D such that D is a closed and bounded subset of \left(\operatorname D, \tau\right), then D is the closed unit ball of \left(X_D, p_D\right) (that is, D = \left\ ) (see footnote for proof).Assume WLOG that X = \operatorname D. Since D is closed in (X, \tau), it is also closed in \left(X_D, p_D\right) and since the seminorm p_D is the
Minkowski functional In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, the ...
of D, which is continuous on \left(X_D, p_D\right), it follows that D is the closed unit ball in \left(X_D, p\right).


Sufficient conditions for a Banach disk

The following theorem may be used to establish that \left(X_D, p_D\right) is a Banach space. Once this is established, D will be a Banach disk in any TVS in which D is bounded. Note that even if D is not a bounded and sequentially complete subset of any Hausdorff TVS, one might still be able to conclude that \left(X_D, p_D\right) is a Banach space by applying this theorem to some disk :K satisfying \left\ \subseteq K \subseteq \left\ because p_D = p_K. The following are consequences of the above theorem: *A sequentially complete bounded disk in a Hausdorff TVS is a Banach disk. *Any disk in a Hausdorff TVS that is complete and bounded (e.g. compact) is a Banach disk. *The closed unit ball in a
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to th ...
is sequentially complete and thus a Banach disk. Suppose that D is a bounded disk in a TVS X. *If L : X \to Y is a continuous linear map and B \subseteq X is a Banach disk, then L(B) is a Banach disk and L\big\vert_ : X_B \to L\left(X_B\right) induces an isometric TVS-isomorphism Y_ \cong X_B / \left(X_B \cap \operatorname L\right).


Properties of Banach disks

Let X be a TVS and let D be a bounded disk in X. If D is a bounded Banach disk in a Hausdorff locally convex space X and if T is a barrel in X then T absorbs D (i.e. there is a number r > 0 such that D \subseteq r T. If U is a convex balanced closed neighborhood of 0 in X then the collection of all neighborhoods r U, where r > 0 ranges over the positive real numbers, induces a topological vector space topology on X. When X has this topology, it is denoted by X_U. Since this topology is not necessarily Hausdorff nor complete, the completion of the Hausdorff space X / p_U^(0) is denoted by \overline so that \overline is a complete Hausdorff space and p_U(x) := \inf_ r is a norm on this space making \overline into a Banach space. The polar of U, U^, is a weakly compact bounded equicontinuous disk in X^ and so is infracomplete. If X is a
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) ...
locally convex 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 ve ...
TVS then for every bounded subset B of X, there exists a bounded disk D in X such that B \subseteq X_D, and both X and X_D induce the same
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
on B.


Induced by a radial disk – quotient

Suppose that X is a topological vector space and V 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 polyto ...
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 ...
and
radial Radial is a geometric term of location which may refer to: Mathematics and Direction * Vector (geometric), a line * Radius, adjective form of * Radial distance, a directional coordinate in a polar coordinate system * Radial set * A bearing from ...
set. Then \left\ is a neighborhood basis at the origin for some locally convex topology \tau_V on X. This TVS topology \tau_V is given by the
Minkowski functional In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, the ...
formed by V, p_V : X \to \R, which is a seminorm on X defined by p_V(x) := \inf_ r. The topology \tau_V is Hausdorff if and only if p_V is a norm, or equivalently, if and only if X / p_V^(0) = \ or equivalently, for which it suffices that V be bounded in X. The topology \tau_V need not be Hausdorff but X / p_V^(0) is Hausdorff. A norm on X / p_V^(0) is given by \left\, x + X / p_V^(0)\right\, := p_V(x), where this value is in fact independent of the representative of the equivalence class x + X / p_V^(0) chosen. The normed space \left(X / p_V^(0), \, \cdot \, \right) is denoted by X_V and its completion is denoted by \overline. If in addition V is bounded in X then the seminorm p_V : X \to \R is a norm so in particular, p_V^(0) = \. In this case, we take X_V to be the vector space X instead of X / \ so that the notation X_V is unambiguous (whether X_V denotes the space induced by a radial disk or the space induced by a bounded disk). The
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
\tau_Q on X / p_V^(0) (inherited from X's original topology) is finer (in general, strictly finer) than the norm topology.


Canonical maps

The canonical map is the
quotient map In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
q_V : X \to X_V = X / p_V^(0), which is continuous when X_V has either the norm topology or the quotient topology. If U and V are radial disks such that U \subseteq Vthen p_U^(0) \subseteq p_V^(0) so there is a continuous linear surjective canonical map q_ : X / p_U^(0) \to X / p_V^(0) = X_V defined by sending x + p_U^(0) \in X_U = X / p_U^(0) to the equivalence class x + p_V^(0), where one may verify that the definition does not depend on the representative of the equivalence class x + p_U^(0) that is chosen. This canonical map has norm \,\leq 1 and it has a unique continuous linear canonical extension to \overline that is denoted by \overline : \overline \to \overline. Suppose that in addition B \neq \varnothing and C are bounded disks in X with B \subseteq C so that X_B \subseteq X_C and the inclusion \operatorname_B^C : X_B \to X_C is a continuous linear map. Let \operatorname_B : X_B \to X, \operatorname_C : X_C \to X, and \operatorname_B^C : X_B \to X_C be the canonical maps. Then \operatorname_C = \operatorname_B^C \circ \operatorname_C : X_B \to X_C and q_V = q_ \circ q_U.


Induced by a bounded radial disk

Suppose that S is a bounded radial disk. Since S is a bounded disk, if D := S then we may create the auxiliary normed space X_D = \operatorname D with norm p_D(x) := \inf_ r; since S is radial, X_S = X. Since S is a radial disk, if V := S then we may create the auxiliary seminormed space X / p_V^(0) with the seminorm p_V(x) := \inf_ r; because S is bounded, this seminorm is a norm and p_V^(0) = \ so X / p_V^(0) = X / \ = X. Thus, in this case the two auxiliary normed spaces produced by these two different methods result in the same normed space.


Duality

Suppose that H is a weakly closed equicontinuous disk in X^ (this implies that H is weakly compact) and let :U := H^ = \left\ be the polar of H. Because U^ = H^ = H by the
bipolar theorem In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a con ...
, it follows that a continuous linear functional f belongs to X^_H = \operatorname H if and only if f belongs to the continuous dual space of \left(X, p_U\right), where p_U is the
Minkowski functional In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, the ...
of U defined by p_U(x) := \inf_ r.


Related concepts

A disk in a TVS is called infrabornivorous if it absorbs all Banach disks. A linear map between two TVSs is called infrabounded if it maps Banach disks to bounded disks.


Fast convergence

A sequence x_ = \left(x_i\right)_^ in a TVS X is said to be fast convergent to a point x \in X if there exists a Banach disk D such that both x and the sequence is (eventually) contained in \operatorname D and x_ \to x in \left(X_D, p_D\right). Every fast convergent sequence is Mackey convergent.


See also

* * * * * * * * * * * *


Notes


References


Bibliography

* * * * * * * * * * * * * *


External links


Nuclear space at ncatlab
{{TopologicalTensorProductsAndNuclearSpaces Functional analysis