Ursescu theorem explained

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

l{R} : X \rightrightarrows Y is a set-valued function and S is a non-empty subset of a topological vector space X :

• the affine span of

S is denoted by \operatorname{aff} S and the linear span is denoted by \operatorname{span} S. S ^{i} : = \operatorname{aint} _{X} S denotes the algebraic interior of S in X. {} ^{i} S: = \operatorname{aint} _{\operatorname{aff} (S - S)} S denotes the relative algebraic interior of S (i.e. the algebraic interior of S in \operatorname{aff}(S - S) ).{} ^{ib} S : = {} ^{i} S if \operatorname{span} \left(S - s _{0\right)} is barreled for some/every s _{0} \in S while {} ^{ib} S : = \varnothing otherwise.

• • If

S is convex then it can be shown that for any x \in X, x \in {} ^{ib} S if and only if the cone generated by S - x is a barreled linear subspace of X or equivalently, if and only if \cup _{n} n (S - x) is a barreled linear subspace of X

• The domain of

l{R} is \operatorname{Dom} l{R} : = \{ x \in X : l{R}(x) ≠ \varnothing \}.

• The image of

l{R} is \operatorname{Im} l{R} : = \cup _{x} l{R}(x). For any subset A \subseteq X, l{R}(A) : = \cup _{x} l{R}(x).

• The graph of

l{R} is \operatorname{gr} l{R} : = \{ (x, y) \in X x Y : y \in l{R}(x) \}. l{R} is closed (respectively, convex) if the graph of l{R} is closed (resp. convex) in X x Y.

• • Note that

l{R} is convex if and only if for all x _{0,} x _{1} \in X and all r \in [ 0 , 1 ], r l{R}\left(x _{0\right)} + ( 1 - r) l{R}\left(x _{1\right)} \subseteq l{R} \left(r x _{0} + ( 1 - r) x _{1\right).}

• The inverse of

l{R} is the set-valued function l{R} ^{-1} : Y \rightrightarrows X defined by l{R} ^{-1} (y) : = \{ x \in X : y \in l{R}(x) \}. For any subset B \subseteq Y, l{R} ^{-1} (B) : = \cup _{y} l{R} ^{-1} (y).

• • If

f : X \to Y is a function, then its inverse is the set-valued function f ^{-1} : Y \rightrightarrows X obtained from canonically identifying f with the set-valued function f : X \rightrightarrows Y defined by x \mapsto \{ f(x)\}. \operatorname{int} _{T} S is the topological interior of S with respect to T, where S \subseteq T. \operatorname{rint} S : = \operatorname{int} _{\operatorname{aff} S} S is the interior of S with respect to \operatorname{aff} S.

Statement

Corollaries

Additional corollaries

The following notation and notions are used for these corollaries, where

l{R} : X \rightrightarrows Y is a set-valued function, S is a non-empty subset of a topological vector space X :

• a convex series with elements of

S is a series of the form $\backslash sum\_^\backslash infty\; r\_i\; s\_i$ where all s _{i} \in S and $\backslash sum\_^\backslash infty\; r\_i\; =\; 1$ is a series of non-negative numbers. If $\backslash sum\_^\backslash infty\; r\_i\; s\_i$ converges then the series is called convergent while if \left(s _{i\right)} ^{inf ty} ^{i =1} is bounded then the series is called bounded and b-convex.S is ideally convex if any convergent b-convex series of elements of S has its sum in S. S is lower ideally convex if there exists a Fréchet space Y such that S is equal to the projection onto X of some ideally convex subset B of X x Y. Every ideally convex set is lower ideally convex.

Related theorems

Robinson–Ursescu theorem

The implication (1)

\implies (2) in the following theorem is known as the Robinson–Ursescu theorem.

References

• • Baggs. Ivan. 1974. Functions with a closed graph. Proceedings of the American Mathematical Society. en. 43. 2. 439–442. 10.1090/S0002-9939-1974-0334132-8. 0002-9939. free.