# Stereotype space

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 cosmos and a *-autonomous category with the standard analytical tools for constructing new spaces, like taking dual spaces, spaces of operators, tensor products, products and coproducts, limits and colimits, and in addition, immediate subspaces, and immediate quotient spaces.

## Definition

A stereotype space is a topological vector space $\displaystyle{ X }$ over the field $\displaystyle{ \mathbb{C} }$ of complex numbers such that the natural map into the second dual space

$\displaystyle{ i:X\to X^{\star\star},\quad i(x)(f)=f(x),\quad x\in X,\quad f\in X^\star }$

is an isomorphism of topological vector spaces (i.e. a linear and a homeomorphic map). Here the dual space $\displaystyle{ X^\star }$ is defined as the space of all linear continuous functionals $\displaystyle{ f:X\to\mathbb{C} }$ endowed with the topology of uniform convergence on totally bounded sets in $\displaystyle{ X }$, and the second dual space $\displaystyle{ X^{\star\star} }$ is the space dual to $\displaystyle{ X^{\star} }$ in the same sense.

The following criterion holds: a topological vector space $\displaystyle{ X }$ is stereotype if and only if it is locally convex and satisfies the following two conditions:

• pseudocompleteness: each totally bounded Cauchy net in $\displaystyle{ X }$ converges,
• pseudosaturateness: each closed convex balanced capacious set $\displaystyle{ D }$ in $\displaystyle{ X }$ is a neighborhood of zero in $\displaystyle{ X }$.

The property of being pseudocomplete is a weakening of the usual notion of completeness, while the property of being pseudosaturated is a weakening of the notion of barreledness of a topological vector space.

## Examples

The class Ste of stereotype spaces is extremely wide, so that it will not be a serious exaggeration to say that all topological vector spaces really used in analysis are stereotype. Each pseudocomplete barreled space $\displaystyle{ X }$ (in particular, each Banach space and each Fréchet space) is stereotype. Its dual space $\displaystyle{ X^\star }$ (which is not barreled, unless $\displaystyle{ X }$ is a Montel space) is stereotype as well. There exist stereotype spaces which are not Mackey spaces.

Some simple connections between the properties of a stereotype space $\displaystyle{ X }$ and those of its dual space $\displaystyle{ X^\star }$ are expressed in the following list of regularities:

• $\displaystyle{ X }$ is a normed space $\displaystyle{ \Longleftrightarrow }$ $\displaystyle{ X }$ is a Banach space $\displaystyle{ \Longleftrightarrow }$ $\displaystyle{ X^\star }$ is a Smith space;
• $\displaystyle{ X }$ is metrizable $\displaystyle{ \Longleftrightarrow }$ $\displaystyle{ X }$ is a Fréchet space $\displaystyle{ \Longleftrightarrow }$ $\displaystyle{ X^\star }$ is a Brauner space;
• $\displaystyle{ X }$ is barreled $\displaystyle{ \Longleftrightarrow }$ $\displaystyle{ X^\star }$ has the Heine-Borel property;
• $\displaystyle{ X }$ is quasi-barreled $\displaystyle{ \Longleftrightarrow }$ in $\displaystyle{ X^\star }$ if a set $\displaystyle{ T }$ is absorbed by each barrel, then $\displaystyle{ T }$ is totally bounded;
• $\displaystyle{ X }$ is a Mackey space $\displaystyle{ \Longleftrightarrow }$ in $\displaystyle{ X^\star }$ every $\displaystyle{ X }$-weakly compact set is compact;
• $\displaystyle{ X }$ is a Montel space $\displaystyle{ \Longleftrightarrow }$ $\displaystyle{ X }$ is barreled and has the Heine-Borel property $\displaystyle{ \Longleftrightarrow }$ $\displaystyle{ X^\star }$ is a Montel space;
• $\displaystyle{ X }$ is a space with a weak topology $\displaystyle{ \Longleftrightarrow }$ in $\displaystyle{ X^\star }$ every compact set $\displaystyle{ T }$ is finite-dimensional;
• $\displaystyle{ X }$ is separable $\displaystyle{ \Longleftrightarrow }$ in $\displaystyle{ X^\star }$ there is a sequence of closed subspaces $\displaystyle{ L_n }$ of finite co-dimension with trivial intersection: $\displaystyle{ \bigcap_{n=1}^{\infty}L_n=\{0\} }$.
• $\displaystyle{ X }$ is complete $\displaystyle{ \Longleftrightarrow }$ $\displaystyle{ X^\star }$ is co-complete $\displaystyle{ \Longleftrightarrow }$ $\displaystyle{ X^\star }$ is saturated;
• $\displaystyle{ X }$ is a Pták space $\displaystyle{ \Longleftrightarrow }$ in $\displaystyle{ X^\star }$ a subspace $\displaystyle{ L }$ is closed if it has the closed intersection $\displaystyle{ L\cap K }$ with each compact set $\displaystyle{ K\subseteq X^\star }$;
• $\displaystyle{ X }$ is hypercomplete $\displaystyle{ \Longleftrightarrow }$ in $\displaystyle{ X^\star }$ an absolutely convex set $\displaystyle{ B }$ is closed if it has the closed intersection $\displaystyle{ B\cap K }$ with each compact set $\displaystyle{ K\subseteq X^\star }$.

Counterexamples:

1. If a metrizable locally convex space $\displaystyle{ X }$ is not complete, then it is not stereotype.

2. If $\displaystyle{ X }$ is an infinite dimensional Banach space, and $\displaystyle{ Y=X'_\sigma }$ is its dual space (of linear continuous functionals $\displaystyle{ f:X\to{\mathbb C} }$) considered with the $\displaystyle{ X }$-weak topology, then $\displaystyle{ Y }$ is not stereotype.

## History

The first results on this type of reflexivity of topological vector spaces were obtained by M. F. Smith in 1952. Further investigations were conducted by B. S. Brudovskii,  W. C. Waterhouse, K. Brauner, S. S. Akbarov, and E. T. Shavgulidze. The term "stereotype space" was introduced by S. S. Akbarov in 1995. The main properties of the category of stereotype spaces were described by S. S. Akbarov in his series of works of 1995-2017.

## Pseudocompletion and pseudosaturation

Each locally convex space $\displaystyle{ X }$ can be transformed into a stereotype space with the help of two standard operations, pseudocompletion and pseudosaturation, defined by the following two propositions.

Theorem. For each locally convex space $\displaystyle{ X }$ there exists a pseudocomplete locally convex space $\displaystyle{ X^\triangledown }$ and a linear continuous mapping $\displaystyle{ {\triangledown}_X :X \to X^{\triangledown} }$ such that for every pseudocomplete locally convex space $\displaystyle{ Y }$ and for every linear continuous mapping $\displaystyle{ \varphi:X\to Y }$ there is a unique linear continuous mapping $\displaystyle{ \varphi^\triangledown:X^\triangledown\to Y }$ such that
$\displaystyle{ \varphi^\triangledown\circ{\triangledown}_X=\varphi }$.

The space $\displaystyle{ X^\triangledown }$ is called a pseudocompletion of the space $\displaystyle{ X }$. It is unique up to an isomorphism of locally convex spaces.

For each linear continuous mapping of locally convex spaces $\displaystyle{ \varphi :X\to Y }$ there is a unique linear continuous mapping $\displaystyle{ \varphi^{\triangledown} :X^{\triangledown} \to Y^{\triangledown} }$ such that

$\displaystyle{ \triangledown_Y\circ\varphi=\varphi^\triangledown\circ\triangledown_X }$,

and the correspondence $\displaystyle{ \varphi\mapsto\varphi^{\triangledown} }$ can be defined as a (covariant) functor.

The pseudocompletion $\displaystyle{ X^{\triangledown} }$ can be defined as an envelope of the locally convex space $\displaystyle{ X }$ in the class $\displaystyle{ \operatorname{PC} }$ of all pseudocomplete locally convex spaces with respect to the same class $\displaystyle{ \operatorname{PC} }$:

$\displaystyle{ X^{\triangledown}=\operatorname{Env}_{\operatorname{PC}}^{\operatorname{PC}} X }$

One can imagine the pseudocompletion of $\displaystyle{ X }$ as the "nearest to $\displaystyle{ X }$ from the outside" pseudocomplete locally convex space, so that the operation $\displaystyle{ X\mapsto X^\triangledown }$ adds to $\displaystyle{ X }$ some supplementary elements, but does not change the topology of $\displaystyle{ X }$ (like the usual operation of completion).

Theorem. For each locally convex space $\displaystyle{ X }$ there is a pseudosaturated locally convex space $\displaystyle{ X^\vartriangle }$ and a linear continuous mapping $\displaystyle{ {\vartriangle}_X :X^{\vartriangle} \to X }$ such that for each pseudosaturated locally convex space $\displaystyle{ Y }$ and for each linear continuous mapping $\displaystyle{ \varphi:Y\to X }$ there is a unique linear continuous mapping $\displaystyle{ \varphi^\vartriangle:Y\to X^\vartriangle }$ such that
$\displaystyle{ {\vartriangle}_X\circ\varphi^\vartriangle=\varphi. }$

The space $\displaystyle{ X^\vartriangle }$ is called a pseudosaturation of the space $\displaystyle{ X }$. It is unique up to an isomorphism of locally convex spaces.

For each linear continuous mapping of locally convex spaces $\displaystyle{ \varphi :Y\to X }$ there is a unique linear continuous mapping $\displaystyle{ \varphi^{\vartriangle} :Y^{\vartriangle} \to X^{\vartriangle} }$ such that

$\displaystyle{ \varphi\ \circ\vartriangle_Y=\vartriangle_X\circ\ \varphi^\vartriangle }$,

and the correspondence $\displaystyle{ \varphi\mapsto\varphi^{\vartriangle} }$ can be defined as a (covariant) functor.

The pseudosaturation $\displaystyle{ X^{\vartriangle} }$ can be defined as a refinement of the locally convex space $\displaystyle{ X }$ in the class $\displaystyle{ \operatorname{PS} }$ of all pseudosaturated locally convex spaces with respect to the same class $\displaystyle{ \operatorname{PS} }$:

$\displaystyle{ X^{\vartriangle}=\operatorname{Ref}_{\operatorname{PS}}^{\operatorname{PS}} X. }$

One can imagine the pseudosaturation of $\displaystyle{ X }$ as the "nearest to $\displaystyle{ X }$ from the inside" pseudosaturated locally convex space, so that the operation $\displaystyle{ X\mapsto X^\vartriangle }$ strengthens the topology of $\displaystyle{ X }$, but does not change the elements of $\displaystyle{ X }$.

If $\displaystyle{ X }$ is a pseudocomplete locally convex space, then its pseudosaturation $\displaystyle{ X^\vartriangle }$ is stereotype. Dually, if $\displaystyle{ X }$ is a pseudosaturated locally convex space, then its pseudocompletion $\displaystyle{ X^\triangledown }$ is stereotype. For arbitrary locally convex space $\displaystyle{ X }$ the spaces $\displaystyle{ X^{\vartriangle\triangledown} }$ and $\displaystyle{ X^{\triangledown\vartriangle} }$ are stereotype.

## Immediate subspaces and immediate quotient spaces

The idea of subspace (and of quotient space) in stereotype theory leads to more complicated results than in the theory of locally convex spaces.

### Immediate subspaces and envelopes

The notion of immediate subspace gives a "concrete description" of the abstract notion of immediate monomorphism, or, what is equivalent in this situation, strong monomorphism in the category Ste. Surprisingly, this description does not coincide with the construction of closed subspace in the category LocConv of locally convex spaces.

• Suppose $\displaystyle{ Y }$ is a subset in a stereotype space $\displaystyle{ X }$ endowed with a structure of a stereotype space in such a way that the set-theoretic inclusion $\displaystyle{ Y\subseteq X }$ is a morphism of stereotype spaces (i.e. a continuous linear map). Then $\displaystyle{ Y }$ is called a subspace of the stereotype space $\displaystyle{ X }$, with the notation
$\displaystyle{ Y\hookrightarrow X }$.
• Suppose we have a chain of stereotype subspaces
$\displaystyle{ Z\hookrightarrow Y\hookrightarrow X }$,
and the first mapping $\displaystyle{ Z\hookrightarrow Y }$ is a bimorphism of stereotype spaces. Then the space $\displaystyle{ Y }$ is called a mediator of the subspace $\displaystyle{ Z }$ in the space $\displaystyle{ X }$.
• A subspace $\displaystyle{ Z }$ in a stereotype space $\displaystyle{ X }$ is called an immediate subspace in $\displaystyle{ X }$, with the notation
$\displaystyle{ Z\overset{\circ}{\hookrightarrow} X }$,
if it has no non-trivial mediators, i.e. for any mediator $\displaystyle{ Y }$ of $\displaystyle{ Z }$ in $\displaystyle{ X }$ the inclusion $\displaystyle{ Z\hookrightarrow Y }$ is an isomorphism.

Examples:

1. An immediate subspace $\displaystyle{ Z }$ in a stereotype space $\displaystyle{ X }$ is said to be closed, if $\displaystyle{ Z }$ (as a set) is closed in $\displaystyle{ X }$ (as a topological space). If $\displaystyle{ Y }$ is a closed subspace in a stereotype space $\displaystyle{ X }$ (as in a locally convex space), then its pseudosaturation $\displaystyle{ Z=Y^\vartriangle }$ is a closed immediate subspace in $\displaystyle{ X }$. All closed immediate subspaces have this form.

2. There are stereotype spaces $\displaystyle{ X }$ with closed immediate subspaces $\displaystyle{ Z=Y^\vartriangle }$ whose topology is not inherited from $\displaystyle{ X }$ (this is one of the qualitative differences with the category LocConv of locally convex spaces).

3. In contrast to the category LocConv of locally convex spaces in the category Ste the immediate subspaces are not always closed.

Theorem. For any set $\displaystyle{ M }$ in a stereotype space $\displaystyle{ X }$ there is a minimal immediate subspace $\displaystyle{ \operatorname{Env}^XM }$ in $\displaystyle{ X }$, containing $\displaystyle{ M }$:
(i) $\displaystyle{ M\subseteq\operatorname{Env}^XM\overset{\circ}{\hookrightarrow} X }$
(ii) $\displaystyle{ \forall Z \quad (M\subseteq Z\overset{\circ}{\hookrightarrow} X\ \Rightarrow\ \operatorname{Env}^XM\subseteq Z) }$,
and this subspace $\displaystyle{ \operatorname{Env}^XM }$ is an immediate subspace in each immediate subspace, containing $\displaystyle{ M }$:
(iii) $\displaystyle{ \forall Z \quad (M\subseteq Z\overset{\circ}{\hookrightarrow} X\ \Rightarrow\ \operatorname{Env}^XM\overset{\circ}{\hookrightarrow} Z) }$,
• The subspace $\displaystyle{ \operatorname{Env}^XM }$ is called an envelope of the set $\displaystyle{ M }$ in the stereotype space $\displaystyle{ X }$.
Theorem. Each set $\displaystyle{ M }$ in a stereotype space $\displaystyle{ X }$ is a total set in its envelope $\displaystyle{ \operatorname{Env}^XM }$.

If $\displaystyle{ {\mathbb C}_M }$ denotes the space of all functions $\displaystyle{ \alpha:M\to{\mathbb C} }$ with finite support, endowed with the strongest locally convex topology, and the mapping $\displaystyle{ \varphi:{\mathbb C}_M\to X }$ acts by the formula $\displaystyle{ \varphi(\alpha)=\sum_{x\in M}\alpha(x)\cdot x }$, then the envelope $\displaystyle{ \operatorname{Env}^XM }$ coincides with the abstract categorical envelope of the space $\displaystyle{ {\mathbb C}_M }$ in the class $\displaystyle{ \operatorname{Epi} }$ of all epimorphisms in the category Ste with respect to the morphism $\displaystyle{ \varphi }$:

$\displaystyle{ \operatorname{Env}^XM=\operatorname{Env}_\varphi^{\operatorname{Epi}}{\mathbb C}_M }$

### Immediate quotient spaces and refinements

Dually, the notion of immediate quotient space gives a "concrete description" of the abstract notion of immediate epimorphism, or, what is equivalent here, strong epimorphism in the category Ste. Like in the situation with monomorphisms, this description does not coincide with the construction of quotient space in the category LocConv of locally convex spaces.

• Let $\displaystyle{ E }$ be a closed subspace (in the usual sense) in a stereotype space $\displaystyle{ X }$. Consider a topology $\displaystyle{ \tau }$ on the quotient space $\displaystyle{ X/E }$, which is majorized by the usual quotient topology of $\displaystyle{ X/E }$. Let $\displaystyle{ (X/E)^\blacktriangledown }$ be a completion of $\displaystyle{ X/E }$ with respect to the topology $\displaystyle{ \tau }$. Suppose $\displaystyle{ Y }$ is a subset in the locally convex space $\displaystyle{ (X/E)^\blacktriangledown }$ which contains $\displaystyle{ X/E }$ and at the same time is a stereotype space. Then $\displaystyle{ Y }$ is called a quotient space of the stereotype space $\displaystyle{ X }$, with the notation
$\displaystyle{ X\rightarrowtail Y }$.
• Suppose we have two quotient spaces $\displaystyle{ X\rightarrowtail Y }$ and $\displaystyle{ X\rightarrowtail Z }$. It is said that the $\displaystyle{ Y }$ subordinates $\displaystyle{ Z }$ (notation: $\displaystyle{ Y\ge Z }$) if there is a morphism $\displaystyle{ \varkappa:Y\to Z }$ such that $\displaystyle{ \upsilon_Z=\varkappa\circ\upsilon_Y }$ (where $\displaystyle{ \upsilon_Z:X\to Z }$ and $\displaystyle{ \upsilon_Y:X\to Y }$ are the natural mappings).
• Suppose that the quotient space $\displaystyle{ X\rightarrowtail Y }$ subordinates the quotient space $\displaystyle{ X\rightarrowtail Z }$ (i.e. $\displaystyle{ Y\ge Z }$) and the corresponding morphism $\displaystyle{ \varkappa:Y\to Z }$ is a bimorphism. Then the quotient space $\displaystyle{ Y }$ is called a mediator of the quotient space $\displaystyle{ Z }$ of the space $\displaystyle{ X }$.
• A quotient space $\displaystyle{ Z }$ of a stereotype space $\displaystyle{ X }$ is called an immediate quotient space of $\displaystyle{ X }$, with the notation
$\displaystyle{ X\overset{\circ}{\rightarrowtail} Z }$,
if it has no non-trivial mediators, i.e. for any mediator $\displaystyle{ Y }$ of $\displaystyle{ Z }$ the morphism $\displaystyle{ \varkappa:Y\to Z }$ is an isomorphism.

Examples:

1. An immediate quotient space $\displaystyle{ Z }$ of a stereotype space $\displaystyle{ X }$ is said to be open, if the corresponding map $\displaystyle{ X\to Z }$ is open. If $\displaystyle{ E }$ is a closed subspace in a stereotype space $\displaystyle{ X }$, then the pseudocompletion $\displaystyle{ Z=(X/E)^\triangledown }$ of the (locally convex) quotient space $\displaystyle{ X/E }$ is an open immediate quotient space of $\displaystyle{ X }$. All open immediate quotient spaces have this form.

2. There are stereotype spaces $\displaystyle{ X }$ with immediate quotient spaces $\displaystyle{ Z }$ which cannot be represented in the form $\displaystyle{ Z=X/E }$.

3. In contrast to the category LocConv of locally convex spaces in the category Ste immediate quotient spaces are not always open.

Theorem. For any set $\displaystyle{ F }$ of linear continuous functionals on a stereotype space $\displaystyle{ X }$ there is a minimal immediate quotient space $\displaystyle{ \operatorname{Ref}^XF }$ of $\displaystyle{ X }$ to which all functionals $\displaystyle{ F }$ can be extended:
(i) $\displaystyle{ X\overset{\circ}{\rightarrowtail}\operatorname{Ref}^XF\ \&\ (\operatorname{Ref}^XF)^\star\supseteq F }$
(ii) $\displaystyle{ \forall Z \quad \big((X\overset{\circ}{\rightarrowtail}Z\ \&\ Z^\star\supseteq F)\ \Rightarrow\ Z\ge\operatorname{Ref}^XF\big) }$,
and this quotient space $\displaystyle{ \operatorname{Ref}^XF }$ is (up to an isomorphism) an immediate quotient space of each immediate quotient space, to which the functionals $\displaystyle{ F }$ are extended:
(iii) $\displaystyle{ \forall Z \quad \big((X\overset{\circ}{\rightarrowtail}Z\ \&\ Z^\star\supseteq F)\ \Rightarrow\ Z\overset{\circ}{\rightarrowtail}\operatorname{Ref}^XF\big) }$.
• The quotient space $\displaystyle{ \operatorname{Ref}^XF }$ is called a refinement of the set $\displaystyle{ F }$ on the stereotype space $\displaystyle{ X }$.
Theorem. Each set $\displaystyle{ F }$ of linear continuous functionals on a stereotype space $\displaystyle{ X }$ is a total set on its refinement $\displaystyle{ \operatorname{Ref}^XF }$.

If $\displaystyle{ {\mathbb C}^F }$ denotes the space of all functions $\displaystyle{ f:M\to{\mathbb C} }$, endowed with the topology of pointwise convergence, and the mapping $\displaystyle{ \varphi:X\to{\mathbb C}^F }$ acts by the formula $\displaystyle{ \varphi(x)(f)=f(x) }$, then the refinement $\displaystyle{ \operatorname{Ref}^XF }$ coincides with the abstract categorical refinement of the space $\displaystyle{ {\mathbb C}^F }$ in the class $\displaystyle{ \operatorname{Mono} }$ of all monomorphisms in the category Ste by means of the morphism $\displaystyle{ \varphi }$:

$\displaystyle{ \operatorname{Ref}^XF=\operatorname{Ref}_\varphi^{\operatorname{Mono}}{\mathbb C}^F }$

## Category "Ste" of stereotype spaces

The class Ste of stereotype spaces forms a category with linear continuous maps as morphisms and possesses the following properties:

### Kernel and cokernel in the category "Ste"

Ste is a pre-abelian category: each morphism $\displaystyle{ \varphi:X\to Y }$ in the category Ste has a kernel

$\displaystyle{ \operatorname{Ker}\varphi=\{x\in X:\varphi(x)=0\}^\vartriangle=\operatorname{Env}^X\{x\in X:\varphi(x)=0\} }$

and a cokernel

$\displaystyle{ \operatorname{Coker}\varphi=\Big(Y\Big/\overline{\varphi(X)}\Big)^\triangledown=\operatorname{Ref}^Y\{f\in Y^\star:\ f\circ\varphi=0\}. }$

As a corollary, $\displaystyle{ \varphi }$ has an image $\displaystyle{ \operatorname{Im}\varphi }$ and a coimage $\displaystyle{ \operatorname{Coim}\varphi }$ as well. The following natural identities hold:

\displaystyle{ \begin{align} & (\operatorname{Ker}\varphi)^\star =\operatorname{Coker}(\varphi^\star), & & (\operatorname{Coker}\varphi)^\star =\operatorname{Ker}(\varphi^\star), \\[5pt] & (\operatorname{Im}\varphi)^\star =\operatorname{Coim}(\varphi^\star), & & (\operatorname{Coim}\varphi)^\star =\operatorname{Im}(\varphi^\star), \\[5pt] & (\operatorname{Ker}\varphi)^{\perp} =\operatorname{Im}(\varphi^\star), & & (\operatorname{Im}\varphi)^{\perp} =\operatorname{Ker}(\varphi^\star), \\[5pt] & \operatorname{Ker}\varphi =(\operatorname{Im}(\varphi^\star))^{\perp}, & & \operatorname{Im}\varphi =(\operatorname{Ker}(\varphi^\star))^{\perp}, \end{align} }

where $\displaystyle{ Q^{\perp} }$ denotes the pseudosaturation of the annihilator of the subspace $\displaystyle{ Q\subseteq P }$ in the dual space $\displaystyle{ P^\star }$:

$\displaystyle{ Q^\perp=\{f\in P^\star: \ f|_Q=0\}^\vartriangle=\operatorname{Env}^{P^\star}\{f\in P^\star: \ f|_Q=0\} }$.

### "Ste" as a *-autonomous category

For any two stereotype spaces $\displaystyle{ X }$ and $\displaystyle{ Y }$ the stereotype space of operators $\displaystyle{ Y\oslash X }$ from $\displaystyle{ X }$ into $\displaystyle{ Y }$, is defined as the pseudosaturation of the space $\displaystyle{ \operatorname{L}(X,Y) }$ of all linear continuous maps $\displaystyle{ \varphi:X\to Y }$ endowed with the topology of uniform convergeance on totally bounded sets. The space $\displaystyle{ Y\oslash X }$ is stereotype. It defines two natural tensor products

$\displaystyle{ X\circledast Y:= (X^\star\oslash Y)^\star, }$
$\displaystyle{ X\odot Y := Y\oslash X^\star. }$
Theorem. In the category Ste the following natural identities hold::
$\displaystyle{ \mathbb{C}\circledast X\cong X\cong X\circledast \mathbb{C}, }$
$\displaystyle{ \mathbb{C}\odot X\cong X\cong X\odot\mathbb{C}, }$
$\displaystyle{ X\circledast Y\cong Y\circledast X, }$
$\displaystyle{ X\odot Y\cong Y\odot X, }$
$\displaystyle{ (X\circledast Y)\circledast Z\cong X\circledast (Y\circledast Z), }$
$\displaystyle{ (X\odot Y)\odot Z\cong X\odot (Y\odot Z), }$
$\displaystyle{ (X\circledast Y)^\star\cong Y^\star\odot X^\star, }$
$\displaystyle{ (X\odot Y)^\star\cong Y^\star\circledast X^\star, }$
$\displaystyle{ X\oslash Y\cong Y^\star\oslash X^\star, }$
$\displaystyle{ X\oslash (Y\circledast Z)\cong (X\oslash Y)\oslash Z, }$
$\displaystyle{ (X\odot Y)\oslash Z\cong X\odot(Y\oslash Z). }$
In particular, Ste is a symmetric monoidal category with respect to the bifunctor $\displaystyle{ \odot }$, a closed symmetric monoidal category with respect to the bifunctor $\displaystyle{ \circledast }$ and the internal hom-functor $\displaystyle{ \oslash }$, and a *-autonomous category:
$\displaystyle{ X^{\star\star}\cong X, }$
$\displaystyle{ X^\star\oslash (Y\circledast Z)\cong (X\circledast Y)^\star\oslash Z. }$

Examples:

1. If $\displaystyle{ X }$ and $\displaystyle{ Y }$ are Fréchet spaces, then their stereotype tensor product $\displaystyle{ X\circledast Y }$ coincides with the usual projective tensor product $\displaystyle{ X\hat{\otimes}Y }$ of locally convex spaces $\displaystyle{ X }$ and $\displaystyle{ Y }$.

2. If $\displaystyle{ X }$ and $\displaystyle{ Y }$ are Fréchet spaces and at least one of them possesses the (classical) approximation property, then their stereotype tensor product $\displaystyle{ X\odot Y }$ coincides with the usual injective tensor product $\displaystyle{ X\check{\otimes}Y }$ of locally convex spaces $\displaystyle{ X }$ and $\displaystyle{ Y }$.

### "Ste" as a cosmos

Ste is a bicomplete category: each small diagram $\displaystyle{ F:J\to\, }$Ste has a colimit (or direct limit), $\displaystyle{ \lim_{j\to\infty}F_j }$, which coincides with the pseudocompletion of the corresponding colimit in the category LocConv of locally convex spaces

$\displaystyle{ \lim_{j\to\infty}F_j=\Big(\overset{\operatorname{\bf LocConv}}{\lim_{j\to\infty}}F_j\Big)^\triangledown }$,

and a limit (or inverse limit), $\displaystyle{ \lim_{\infty\gets j}F_j }$, which coincides with the pseudosaturation of the corresponding limit in LocConv

$\displaystyle{ \lim_{\infty\gets j}F_j=\Big(\overset{\operatorname{\bf LocConv}}{\lim_{\infty\gets j}}F_j\Big)^\vartriangle }$.

However, the direct sum and the direct product in Ste coincide with the corresponding constructions in LocConv:

$\displaystyle{ \bigoplus_{i\in I} X_i=\overset{\operatorname{\bf LocConv}}{\bigoplus_{i\in I}} X_i,\quad \prod_{i\in I} X_i=\overset{\operatorname{\bf LocConv}}{\prod_{i\in I}X_i}. }$

Together with the symmetric closed monoidal structure, the existence of limits and colimits implies the following property:

Theorem. The category Ste is a cosmos.

The following natural identities hold:

$\displaystyle{ \Big(\bigoplus_{i\in I} X_i\Big)^\star\cong \prod_{i\in I} X_i^\star }$
$\displaystyle{ \Big(\prod_{i\in I} X_i\Big)^\star\cong \bigoplus_{i\in I} X_i^\star }$
$\displaystyle{ Y\oslash \Big(\bigoplus_{i\in I} X_i\Big)\cong \prod_{i\in I} ( Y\oslash X_i) }$
$\displaystyle{ \Big(\prod_{j\in J} Y_j\Big)\oslash X\cong \prod_{j\in J} ( Y_j\oslash X) }$
$\displaystyle{ \Big(\bigoplus_{i\in I} X_i\Big)\circledast \Big(\bigoplus_{j\in J} Y_j\Big) \cong \bigoplus_{i\in I,j\in J} ( X_i\circledast Y_j) }$
$\displaystyle{ \Big(\prod_{i\in I} X_i\Big)\odot \Big(\prod_{j\in J} Y_j\Big) \cong \prod_{i\in I,j\in J} ( X_i\odot Y_j) }$
$\displaystyle{ \Big(\lim_{i\to\infty} X_i\Big)^\star\cong \lim_{\infty\gets i} X_i^\star }$
$\displaystyle{ \Big(\lim_{\infty\gets i} X_i\Big)^\star\cong \lim_{i\to\infty} X_i^\star }$
$\displaystyle{ Y\oslash \Big(\lim_{i\to\infty} X_i\Big)\cong \lim_{\infty\gets i} (Y\oslash X_i) }$
$\displaystyle{ \Big(\lim_{\infty\gets j} Y_j\Big)\oslash X\cong \lim_{\infty\gets j} (Y_j\oslash X) }$
$\displaystyle{ \Big(\lim_{i\to\infty} X_i\Big)\circledast \Big(\lim_{j\to\infty} Y_j\Big) \cong \lim_{i,j\to\infty} ( X_i\circledast Y_j) }$
$\displaystyle{ \Big(\lim_{\infty\gets i} X_i\Big)\odot \Big(\lim_{\infty\gets j} Y_j\Big) \cong \lim_{\infty\gets i,j} ( X_i\odot Y_j) }$

### Grothendieck transformation

If $\displaystyle{ X }$ and $\displaystyle{ Y }$ are stereotype spaces then for each elements $\displaystyle{ x\in X }$ and $\displaystyle{ y\in Y }$ the formula

$\displaystyle{ (x\circledast y)(\varphi)=\varphi(y)(x),\qquad \varphi\in X^\star\oslash Y }$

defines an elementary tensor $\displaystyle{ x\circledast y\in X\circledast Y=(X^\star\oslash Y)^\star }$, and the formula

$\displaystyle{ (x\odot y)(f)=f(x)\cdot y,\qquad f\in X^\star }$

defines an elementary tensor $\displaystyle{ x\odot y\in X\odot Y=Y\oslash X^\star }$

Theorem. For each stereotype spaces $\displaystyle{ X }$ and $\displaystyle{ Y }$ there is a unique linear continuous map $\displaystyle{ \Gamma_{X,Y}: X\circledast Y\to X\odot Y }$ which turns elementary tensors $\displaystyle{ x\circledast y }$ into elementary tensors $\displaystyle{ x\odot y }$:
$\displaystyle{ \Gamma_{X,Y}(x\circledast y)=x\odot y,\qquad x\in X, \ y\in Y. }$
The family of maps $\displaystyle{ \Gamma_{X,Y}: X\circledast Y\to X\odot Y }$ defines a natural transformation of the bifunctor $\displaystyle{ \circledast }$ into the bifunctor $\displaystyle{ \odot }$.
• The map $\displaystyle{ \Gamma_{X,Y} }$ is called the Grothendieck transformation.

### Stereotype approximation property

A stereotype space $\displaystyle{ X }$ is said to have the stereotype approximation property, if each linear continuous map $\displaystyle{ \varphi:X\to X }$ can be approximated in the stereotype space of operators $\displaystyle{ X\oslash X }$ by the linear continuous maps of finite rank. This condition is weaker than the existence of the Schauder basis, but formally stronger than the classical approximation property (however, it is not clear (2017) whether the stereotype approximation property coincides with the classical one, or not).

Theorem. For a stereotype space $\displaystyle{ X }$ the following conditions are equivalent:
(i) $\displaystyle{ X }$ has the stereotype approximation property;
(ii) the Grothendieck transformation $\displaystyle{ \Gamma_{X,X^\star}:X\circledast X^\star\to X\odot X^\star }$ is a monomorphism (in the category Ste);
(iii) the Grothendieck transformation $\displaystyle{ \Gamma_{X,X^\star}:X\circledast X^\star\to X\odot X^\star }$ is an epimorphism (in the category Ste);
(iv) for any stereotype space $\displaystyle{ Y }$ the Grothendieck transformation $\displaystyle{ \Gamma_{X,Y}:X\circledast Y\to X\odot Y }$ is a monomorphism (in the category Ste);
(v) for any stereotype space $\displaystyle{ Y }$ the Grothendieck transformation $\displaystyle{ \Gamma_{X,Y}:X\circledast Y\to X\odot Y }$ is an epimorphism (in the category Ste).
Theorem. If two stereotype spaces $\displaystyle{ X }$ and $\displaystyle{ Y }$ have the stereotype approximation property, then the spaces $\displaystyle{ X\oslash Y }$, $\displaystyle{ X\circledast Y }$ and $\displaystyle{ X\odot Y }$ have the stereotype approximation property as well.

In particular, if $\displaystyle{ X }$ has the stereotype approximation property, then the same is true for $\displaystyle{ X^\star }$ and for $\displaystyle{ X\oslash X }$.

### Universality of tensor product

For any stereotype spaces $\displaystyle{ X }$, $\displaystyle{ Y }$, $\displaystyle{ Z }$ a bilinear map $\displaystyle{ \beta:X\times Y\to Z }$ is said to be continuous (as a bilinear map of stereotype spaces) if

1) for each neighborhood of zero $\displaystyle{ W\subseteq Z }$ and for each compact set $\displaystyle{ S\subseteq X }$ there exists a neighborhood of zero $\displaystyle{ V\subseteq Y }$ such that $\displaystyle{ \beta(S,V)\subseteq W }$, and
2) for each neighborhood of zero $\displaystyle{ W\subseteq Z }$ and for each compact set $\displaystyle{ T\subseteq Y }$ there exists a neighborhood of zero $\displaystyle{ U\subseteq X }$ such that $\displaystyle{ \beta(U,T)\subseteq W }$.

Examples:

1. For any stereotype space $\displaystyle{ X }$ the pairing $\displaystyle{ (x,f)\in X\times X^\star\to f(x)\in {\mathbb C} }$ is a continuous bilinear map.

2. For any two stereotype spaces $\displaystyle{ X }$ and $\displaystyle{ Y }$ the map $\displaystyle{ (x,y)\in X\times Y\to x\circledast y\in X\circledast Y }$ is a continuous bilinear map.

3. For any two stereotype spaces $\displaystyle{ X }$ and $\displaystyle{ Y }$ the map $\displaystyle{ (x,y)\in X\times Y\to x\odot y\in X\odot Y }$ is a continuous bilinear map.

Theorem. For any stereotype spaces $\displaystyle{ X }$, $\displaystyle{ Y }$, $\displaystyle{ Z }$ and for any continuous bilinear map $\displaystyle{ \beta:X\times Y\to Z }$ there exists a unique continuous linear map $\displaystyle{ \widetilde{\beta}:X\circledast Y\to Z }$ such that $\displaystyle{ \widetilde{\beta}\circ\iota=\beta }$, where $\displaystyle{ \iota(x,y)=x\circledast y, \ x\in X, \ y\in Y }$.
Corollary. For any stereotype space $\displaystyle{ X }$ the pairing $\displaystyle{ (x,f)\in X\times X^\star\to f(x)\in {\mathbb C} }$ has a unique extension to a linear continuous functional $\displaystyle{ \operatorname{cont}:X\circledast X^\star\to{\mathbb C} }$. This functional in its turn can be represented as a trace of the operators $\displaystyle{ \varphi:X\to X }$ occurring as images of the tensors $\displaystyle{ \alpha\in X\circledast X^\star }$ under the Grothendieck transformation $\displaystyle{ \Gamma_{X,X^\star}: X\circledast X^\star\to X\odot X^\star=X\oslash X }$ if and only if the space $\displaystyle{ X }$ has the stereotype approximation property.

## Applications

Being a symmetric monoidal category, Ste generates the notions of a stereotype algebra (as a monoid in Ste) and a stereotype module (as a module in Ste over such a monoid), and it turns out that for each stereotype algebra $\displaystyle{ A }$ the categories $\displaystyle{ A }$Ste and Ste$\displaystyle{ A }$ of left and right stereotype modules over $\displaystyle{ A }$ have the structure of enriched categories over Ste. This distinguishes the category Ste from the other known categories of locally convex spaces since up to the recent time only the category Ban of Banach spaces and the category Fin of finite-dimensional spaces had been known to possess this property. On the other hand, the category Ste is so wide, and the tools for creating new spaces in Ste are so diverse, that this suggests the idea that all the results of functional analysis can be reformulated inside the stereotype theory without essential losses. On this way one can even try to completely replace the category of locally convex spaces in analysis (and in related areas) by the category Ste of stereotype spaces with the view of possible simplifications – this program was announced by S. Akbarov in 2005 and the following results can be considered as evidence of its reasonableness:

• In the theory of stereotype spaces, the approximation property is inherited by the spaces of operators and by tensor products. This allows to reduce the list of counterexamples in comparison with the Banach theory, where as is known the space of operators does not inherit the approximation property.
• The arising theory of stereotype algebras allows to simplify constructions in the duality theories for non-commutative groups. In particular, the group algebras (and their envelopes in the necessary cases) in these theories become Hopf algebras in the standard algebraic sense.
• This in its turn leads to a family of generalizations of the Pontryagin duality based on the notion of envelope: the holomorphic, the smooth and the continuous envelopes of stereotype algebras give rise respectively to the holomorphic, the smooth and the continuous dualities in big geometric disciplinescomplex geometry, differential geometry, and topology – for certain classes of (not necessarily commutative) topological groups considered in these disciplines (affine algebraic groups, and some classes of Lie groups and Moore groups).