nLab
stereotype space

Context

Functional analysis

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

In functional analysis and related areas of mathematics stereotype spaces are topological vector spaces defined by a special variant of reflexivity condition. They form a class of spaces with a series of remarkable properties, in particular, this class is very wide (for instance, it contains all Fréchet spaces and thus, all Banach spaces), it consists of spaces satisfying a natural condition of completeness, and it forms a closed monoidal category with the standard analytical tools for constructing new spaces, like taking closed subspace, quotient space, projective and injective limits, the space of operators, tensor products, etc.

Definition

Let XX be a topological vector space over the complex numbers \mathbb{C}. Let X *X^\ast be the vector space of continuous linear functionals XX \to \mathbb{C} endowed with the topology of uniform convergence on totally bounded subsets of XX1. This X *X^\ast is a topological vector space, and may be regarded as a form of dual space of XX. This construction defines a contravariant functor () *(-)^\ast on the category of topological vector spaces.

There is a canonical natural transformation i X:XX **i_X: X \to X^{\ast\ast}, sending xXx \in X to the functional defined by i(x)(f)f(x)i(x)(f) \coloneqq f(x).

Definition

A TVS XX is a stereotype space if i Xi_X is an isomorphism.

There is another way of characterizing stereotype spaces XX directly in terms of the TVS structure on XX. We say that a TVS XX is pseudo-complete if every totally bounded Cauchy net in XX converges. Next, call a subset CC of XX capacious if, for every totally bounded set AXA \subset X, there exists a finite set of elements x 1,,x nx_1, \ldots, x_n such that every yA¯y \in \bar{A} is contained in some translate x i+Cx_i + C. Then, say that XX is pseudo-saturated if every closed convex balanced capacious subset of 00 is a neighborhood of 00 (this can be regarded as a weakening of the notion of barreledness).

Theorem

A TVS XX is a stereotype space if and only if it is locally convex, pseudo-complete, and pseudo-saturated.

Basic results

Proposition

The inclusion of pseudo-saturated locally convex TVS, as a full subcategory of locally convex TVS, is coreflective. The right adjoint to the inclusion takes XX to a space denoted X X^\triangle, called the pseudo-saturation of XX. If XX is pseudo-complete, so is X X^\triangle.

Let XX, YY be stereotype spaces, and let L(X,Y)L(X, Y) be the space of continuous linear maps XYX \to Y. Then L(X,Y)L(X, Y) is locally convex, and as YY is pseudo-complete, it follows that L(X,Y)L(X, Y) is also pseudo-complete. However, L(X,Y)L(X, Y) might not be pseudo-saturated. Let Hom(X,Y)Hom(X, Y) be the pseudo-saturation of L(X,Y)L(X, Y), so that Hom(X,Y)Hom(X, Y) is a stereotype space. This construction endows the category of stereotype spaces with a closed category structure, with unit \mathbb{C}. We have X *Hom(X,)X^\ast \cong Hom(X, \mathbb{C}).

Theorem

The category of stereotype spaces, as a closed category with duality () *(-)^\ast, acquires a structure of complete and cocomplete *\ast-autonomous category.

In slightly more detail, a tensor product on stereotype spaces may be defined by the formula XYHom(X,Y *) *X \otimes Y \coloneqq Hom(X, Y^\ast)^\ast. Then for stereotype spaces X,Y,ZX, Y, Z there is a natural isomorphism

hom(XY,Z)hom(X,Hom(Y,Z))\hom(X \otimes Y, Z) \cong \hom(X, Hom(Y, Z))

and there are associativity, symmetry, and unit constraints on \otimes so that in this way, stereotype spaces form a symmetric monoidal closed category. By taking D=I=D = I = \mathbb{C} as dualizing object, it is moreover *\ast-autonomous. It is not, however, autonomous in the sense of compact closed categories, i.e., we do not have an isomorphism between (XY) *(X \otimes Y)^\ast and X *Y *X^\ast \otimes Y^\ast. Rather, we get a second tensor product via the formula

XY(X *Y *) *X \odot Y \coloneqq (X^\ast \otimes Y^\ast)^\ast

and this tensor product plays the role of multiplicative disjunction in linear logic (where we regard *\ast-autonomous categories as models of multiplicative linear logic).

References

  • S.S. Akbarov, Pontryagin duality in the theory of topological vector spaces and in topological algebra, Journal of Mathematical Sciences 113 (2) (2203), 179–349.

  1. In other words, the smallest topology on X *X^\ast that includes all sets L(B,U){f:X:f(B)U}L(B, U) \coloneqq \{f: X \to \mathbb{C}: f(B) \subseteq U\} where BB is totally bounded in XX and UU is open in \mathbb{C}.

Last revised on November 10, 2013 at 11:08:00. See the history of this page for a list of all contributions to it.