In mathematics, F. Riesz's theorem (named after

Frigyes Riesz Frigyes Riesz (, , sometimes known in English and French as Frederic Riesz; 22 January 1880 – 28 February 1956) was a HungarianEberhard Zeidler: Nonlinear Functional Analysis and Its Applications: Linear monotone operators. Springer, 199/ref> ...

) is an important theorem in

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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...

that states that a Hausdorff

topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

(TVS) is finite-dimensional if and only if it is

locally compact 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 ...

. The theorem and its consequences are used ubiquitously in functional analysis, often used without being explicitly mentioned.

Statement

Recall that a

(TVS)

X

is Hausdorff if and only if the singleton set

\

consisting entirely of the origin is a closed subset of

X.

A map between two TVSs is called a TVS-isomorphism or an isomorphism in the category of TVSs if it is a

linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...

homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

Consequences

Throughout,

F, X, Y

are TVSs (not necessarily Hausdorff) with

F

a finite-dimensional vector space. * Every finite-dimensional vector subspace of a Hausdorff TVS is a closed subspace. * All finite-dimensional Hausdorff TVSs are

Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...

s and all norms on such a space are equivalent. * Closed + finite-dimensional is closed: If

M

is a closed vector subspace of a TVS

Y

and if

F

is a finite-dimensional vector subspace of

Y

(

Y, M,

and

F

are not necessarily Hausdorff) then

M + F

is a closed vector subspace of

Y.

* Every vector space isomorphism (i.e. a

bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

) between two finite-dimensional Hausdorff TVSs is a TVS isomorphism. * Uniqueness of topology: If

X

is a finite-dimensional vector space and if

\tau_1

and

\tau_2

are two Hausdorff TVS topologies on

X

then

\tau_1 = \tau_2.

* Finite-dimensional domain: A

linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that p ...

L : F \to Y

between Hausdorff TVSs is necessarily continuous. ** In particular, every

linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...

of a finite-dimensional Hausdorff TVS is continuous. * Finite-dimensional range: Any continuous

surjective In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...

L : X \to Y

with a Hausdorff finite-dimensional range is an

open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...

and thus a topological homomorphism. In particular, the range of

L

is TVS-isomorphic to

X / L^(0).

* A TVS

X

(not necessarily Hausdorff) is locally compact if and only if

X / \overline

is finite dimensional. * The

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

of a compact subset of a finite-dimensional Hausdorff TVS is compact. ** This implies, in particular, that the convex hull of a compact set is equal to the convex hull of that set. * A Hausdorff

locally bounded In mathematics, a function (mathematics), function is locally bounded if it is bounded function, bounded around every point. A Family (disambiguation)#Mathematics, family of functions is locally bounded if for any point in their domain of a functio ...

TVS with the Heine-Borel property is necessarily finite-dimensional.

References

Bibliography

* * * * {{Functional analysis Theorems in functional analysis Lemmas Topological vector spaces

Statement

Consequences

See also

References

Bibliography