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 o ...

and related areas of mathematics a Brauner space is a complete

compactly generated In mathematics, compactly generated can refer to: * Compactly generated group, a topological group which is algebraically generated by one of its compact subsets *Compactly generated space In topology, a compactly generated space is a topological s ...

locally convex 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 v ...

X

having a sequence of compact sets

K_n

such that every other compact set

T\subseteq X

is contained in some

K_n

. Brauner spaces are named afte
Kalman George Brauner
who began their study. All Brauner spaces are stereotype and are in the stereotype duality relations with

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 ...

s: :* for any Fréchet space

X

its stereotype dual space

X^\star

is a Brauner space, :* and vice versa, for any Brauner space

X

its stereotype dual space

X^\star

is a Fréchet space. Special cases of Brauner spaces are Smith spaces.

Examples

* Let

M

be a

\sigma

-compact

locally compact topological space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...

, and

(M)

the

of all continuous functions on

M

(with values in

), endowed with the usual topology of uniform convergence on compact sets in

M

. The dual space

^\star(M)

Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...

s with compact support on

M

with the topology of uniform convergence on compact sets in

(M)

is a Brauner space. * Let

M

be a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

, and

(M)

the

of all smooth functions on

M

(with values in

), endowed with the usual topology of uniform convergence with each derivative on compact sets in

M

. The dual space

^\star(M)

of distributions with compact support in

M

with the topology of uniform convergence on bounded sets in

(M)

is a Brauner space. * Let

M

be a

Stein manifold In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a Ste ...

and

(M)

the

of all holomorphic functions on

M

with the usual topology of uniform convergence on compact sets in

M

. The dual space

^\star(M)

of analytic functionals on

M

with the topology of uniform convergence on bounded sets in

(M)

is a Brauner space. In the special case when

M=G

possesses a structure of a

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...

the spaces

^\star(G)

^\star(G)

^\star(G)

become natural examples of stereotype group algebras. * Let

M\subseteq^n

be a complex

affine algebraic variety Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine com ...

. The space

(M)=_1,...,x_n \

of polynomials (or regular functions) on

M

, being endowed with the strongest locally convex topology, becomes a Brauner space. Its stereotype dual space

^\star(M)

(of currents on

M

) is a

. In the special case when

M=G

is an

affine algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n wh ...

^\star(G)

becomes an example of a stereotype group algebra. * Let

G

be a compactly generated Stein group.I.e. a

which is at the same time a

. The space

_(G)

of all holomorphic functions of exponential type on

G

is a Brauner space with respect to a natural topology.

Notes

References

* * * {{Topological vector spaces Functional analysis Topological vector spaces

Examples

See also

Notes

References