nLab stereotype space

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.

• From Wikipedia: Stereotype space

Definition

Let $X$ be a topological vector space over the complex numbers $\mathbb{C}$. Let $X^\ast$ be the vector space of continuous linear functionals $X \to \mathbb{C}$ endowed with the topology of uniform convergence on totally bounded subsets of $X$¹. This $X^\ast$ is a topological vector space, and may be regarded as a form of dual space of $X$. This construction defines a contravariant functor $(-)^\ast$ on the category of topological vector spaces.

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

Definition

A TVS $X$ is a stereotype space if $i_X$ is an isomorphism.

There is another way of characterizing stereotype spaces $X$ directly in terms of the TVS structure on $X$. We say that a TVS $X$ is pseudo-complete if every totally bounded Cauchy net in $X$ converges. Next, call a subset $C$ of $X$ capacious if, for every totally bounded set $A \subset X$, there exists a finite set of elements $x_1, \ldots, x_n$ such that every $y \in \bar{A}$ is contained in some translate $x_i + C$. Then, say that $X$ is pseudo-saturated if every closed convex balanced capacious subset of $0$ is a neighborhood of $0$ (this can be regarded as a weakening of the notion of barreledness).

Theorem

A TVS $X$ 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 $X$ to a space denoted $X^\triangle$, called the pseudo-saturation of $X$. If $X$ is pseudo-complete, so is $X^\triangle$.

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

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 $X \otimes Y \coloneqq Hom(X, Y^\ast)^\ast$. Then for stereotype spaces $X, Y, Z$ there is a natural isomorphism

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

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 $D = I = \mathbb{C}$ 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 $(X \otimes Y)^\ast$ and $X^\ast \otimes Y^\ast$. Rather, we get a second tensor product via the formula

$$X \odot Y \coloneqq (X^\ast \otimes Y^\ast)^\ast$$

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.

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