Stereotype Space

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The content is sourced from: https://handwiki.org/wiki/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.

functional analysis natural condition reflexivity

References

Akbarov 2003, p. 219.
...or over the field [math]\displaystyle{ \mathbb }[/math] of real numbers, with the similar definition.
Akbarov 2003, p. 219, 220.
A set [math]\displaystyle{ D\subseteq X }[/math] is said to be capacious if for each totally bounded set [math]\displaystyle{ A\subseteq X }[/math] there is a finite set [math]\displaystyle{ F\subseteq X }[/math] such that [math]\displaystyle{ A\subseteq D+F }[/math].
Akbarov 2005.
Akbarov 2003, p. 220, Example 4.3.
Of course, this is a general fact: if [math]\displaystyle{ X }[/math] is stereotype then [math]\displaystyle{ X^\star }[/math] is also stereotype.
Akbarov 2003, p. 221, Example 4.8.
Akbarov 2003, p. 221, Theorem 4.11.
A locally convex space [math]\displaystyle{ X }[/math] is called co-complete if each linear functional [math]\displaystyle{ f:X\to\mathbb }[/math] which is continuous on every totally bounded set [math]\displaystyle{ S\subseteq X }[/math], is automatically continuous on the whole space [math]\displaystyle{ X }[/math].
A locally convex space [math]\displaystyle{ X }[/math] is said to be saturated if for an absolutely convex set [math]\displaystyle{ B\subseteq X }[/math] being a neighbourhood of zero in [math]\displaystyle{ X }[/math] is equivalent to the following: for each totally bounded set [math]\displaystyle{ S\subseteq X }[/math] there is a closed neighbourhood of zero [math]\displaystyle{ U }[/math] in [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ B\cap S=U }[/math].
A locally convex space [math]\displaystyle{ X }[/math] is called a Pták space, or a fully complete space, if in its dual space [math]\displaystyle{ X^\star }[/math] a subspace [math]\displaystyle{ Q\subseteq X^\star }[/math] is [math]\displaystyle{ X }[/math]-weakly closed when it has [math]\displaystyle{ X }[/math]-weakly closed intersection with the polar [math]\displaystyle{ U^\circ }[/math] of each neighbourhood of zero [math]\displaystyle{ U\subseteq X }[/math].
A locally convex space [math]\displaystyle{ X }[/math] is said to be hypercomplete if in its dual space [math]\displaystyle{ X^\star }[/math] every absolutely convex space [math]\displaystyle{ Q\subseteq X^\star }[/math] is [math]\displaystyle{ X }[/math]-weakly closed if it has [math]\displaystyle{ X }[/math]-weakly closed intersection with the polar [math]\displaystyle{ U^\circ }[/math] of each neighbourhood of zero [math]\displaystyle{ U\subseteq X }[/math].
Akbarov 2003, p. 221, Example 4.10.
Akbarov 2003, p. 221, Example 4.9.
Smith 1952.
Onishchik 1984.
Brudovski 1967.
Waterhouse 1968.
Brauner 1973.
Akbarov 2003.
Akbarov 2009.
Akbarov 2016.
Akbarov 2017.
Akbarov & Shavgulidze 2003.
Akbarov 1995.
Akbarov 2003, p. 197.
Akbarov 2022, Theorem 3.3.10.
Akbarov 2003, p. 200.
Akbarov 2022, Theorem 3.3.24.
It is not clear (2017) whether [math]\displaystyle{ X^{\vartriangle\triangledown} }[/math] and [math]\displaystyle{ X^{\triangledown\vartriangle} }[/math] coincide.
A monomorphism [math]\displaystyle{ \mu }[/math] is said to be immediate if in each representation [math]\displaystyle{ \mu=\mu'\circ\varepsilon }[/math], where [math]\displaystyle{ \mu' }[/math] is a monomorphism and [math]\displaystyle{ \varepsilon }[/math] is an epimorphism, the morphism [math]\displaystyle{ \varepsilon }[/math] is automatically an isomorphism.
Akbarov 2016, p. 39.
A monomorphism [math]\displaystyle{ \mu:C\to D }[/math] is said to be strong, if for any epimorphism [math]\displaystyle{ \varepsilon:A\to B }[/math] and for any morphisms [math]\displaystyle{ \alpha:A\to C }[/math] and [math]\displaystyle{ \beta:B\to D }[/math] such that [math]\displaystyle{ \beta\circ\varepsilon=\mu\circ\alpha }[/math] there exists a morphism [math]\displaystyle{ \delta:B\to C }[/math], such that [math]\displaystyle{ \delta\circ\varepsilon=\alpha }[/math] and [math]\displaystyle{ \mu\circ\delta=\beta }[/math].
In other words, in this case the topology of [math]\displaystyle{ Y }[/math] inherited from [math]\displaystyle{ X }[/math] is not pseudosaturated.
Akbarov 2016, p. 128.
Akbarov 2016, p. 134.
Akbarov 2022, Theorem 4.3.17.
I.e. the linear span of [math]\displaystyle{ M }[/math] is dense in [math]\displaystyle{ \operatorname^XM }[/math] (as in a locally convex space).
Akbarov 2016, p. 144.
An epimorphism [math]\displaystyle{ \varepsilon }[/math] is said to be immediate if in each representation [math]\displaystyle{ \varepsilon=\mu\circ\varepsilon' }[/math], where [math]\displaystyle{ \mu }[/math] is a monomorphism and [math]\displaystyle{ \varepsilon' }[/math] is an epimorphism, the morphism [math]\displaystyle{ \mu }[/math] is automatically an isomorphism.
An epimorphism [math]\displaystyle{ \varepsilon:A\to B }[/math] is said to be strong, if for any monomorphism [math]\displaystyle{ \mu:C\to D }[/math] and for any morphisms [math]\displaystyle{ \alpha:A\to C }[/math] and [math]\displaystyle{ \beta:B\to D }[/math] such that [math]\displaystyle{ \beta\circ\varepsilon=\mu\circ\alpha }[/math] there exists a morphism [math]\displaystyle{ \delta:B\to C }[/math], such that [math]\displaystyle{ \delta\circ\varepsilon=\alpha }[/math] and [math]\displaystyle{ \mu\circ\delta=\beta }[/math].
A linear map [math]\displaystyle{ \varphi:X\to Y }[/math] is said to be open, if for each neighborhood of zero [math]\displaystyle{ U\subseteq X }[/math] there is a neighborhood of zero [math]\displaystyle{ V\subseteq Y }[/math] such that [math]\displaystyle{ \varphi(U)\supseteq V\cap\varphi(X) }[/math].
Akbarov 2016, p. 138.
Akbarov 2016, p. 140.
Akbarov 2022, Theorem 4.3.42.
I.e. if [math]\displaystyle{ y\ne z\in\operatorname^XF }[/math], then there exists [math]\displaystyle{ f\in F }[/math] such that [math]\displaystyle{ f(y)\ne f(z) }[/math].
Akbarov 2003, p. 224.
Akbarov 2003, p. 226.
Akbarov 2003, p. 220.
Akbarov 2016, p. 142.
Akbarov 2003, p. 245.
Akbarov 2009, p. 480-481.
Akbarov 2017, p. 581.
Akbarov 2003, 7.17.
Akbarov 2003, 7.21.
Akbarov 2003, (4.15).
Akbarov 2003, p. 246.
Akbarov 2003, p. 264.
Akbarov 2003, p. 265.
Akbarov 2003, p. 242.
Akbarov 2003, p. 289.
Szankowski 1981.
Akbarov 2003, p. 278.
Akbarov 2009, p. 507.
Kuznetsova 2013.

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