Ursescu theorem

In mathematics, particularly in functional analysis and convex analysis, the Ursescu theorem is a theorem that generalizes the closed graph theorem, the open mapping theorem, and the uniform boundedness principle.

Ursescu theorem

The following notation and notions are used, where is a set-valued function and is a non-empty subset of a topological vector space :

• the affine span of is denoted by and the linear span is denoted by
• denotes the algebraic interior of in
• denotes the relative algebraic interior of (i.e. the algebraic interior of in ).
• if is barreled for some/every while otherwise.
• If is convex then it can be shown that for any if and only if the cone generated by is a barreled linear subspace of or equivalently, if and only if is a barreled linear subspace of
• The domain of is
• The image of is For any subset
• The graph of is
• is closed (respectively, convex) if the graph of is closed (resp. convex) in
• Note that is convex if and only if for all and all
• The inverse of is the set-valued function defined by For any subset
• If is a function, then its inverse is the set-valued function obtained from canonically identifying with the set-valued function defined by
• is the topological interior of with respect to where
• is the interior of with respect to

Statement

Corollaries

Closed graph theorem

Uniform boundedness principle

Open mapping theorem

Additional corollaries

The following notation and notions are used for these corollaries, where is a set-valued function, is a non-empty subset of a topological vector space :

• a convex series with elements of is a series of the form where all and is a series of non-negative numbers. If converges then the series is called convergent while if is bounded then the series is called bounded and b-convex.
• is ideally convex if any convergent b-convex series of elements of has its sum in
• is lower ideally convex if there exists a Fréchet space such that is equal to the projection onto of some ideally convex subset B of Every ideally convex set is lower ideally convex.

Related theorems

Simons' theorem

Robinson–Ursescu theorem

The implication (1) (2) in the following theorem is known as the Robinson–Ursescu theorem.

See also

Notes

References