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 $$\mathcal{R} : X \rightrightarrows Y$$ 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 $$\cup_{n \in \N} n (S - x)$$ is a barreled linear subspace of $$X$$
 * The domain of $$\mathcal{R}$$ is $$\operatorname{Dom} \mathcal{R} := \{ x \in X : \mathcal{R}(x) \neq \varnothing \}.$$
 * The image of $$\mathcal{R}$$ is $$\operatorname{Im} \mathcal{R} := \cup_{x \in X} \mathcal{R}(x).$$ For any subset $$A \subseteq X,$$ $$\mathcal{R}(A) := \cup_{x \in A} \mathcal{R}(x).$$
 * The graph of $$\mathcal{R}$$ is $$\operatorname{gr} \mathcal{R} := \{ (x, y) \in X \times Y : y \in \mathcal{R}(x) \}.$$
 * $$\mathcal{R}$$ is closed (respectively, convex) if the graph of $$\mathcal{R}$$ is closed (resp. convex) in $$X \times Y.$$
 * Note that $$\mathcal{R}$$ is convex if and only if for all $$x_0, x_1 \in X$$ and all $$r \in [0, 1],$$ $$r \mathcal{R}\left(x_0\right) + (1 - r) \mathcal{R}\left(x_1\right) \subseteq \mathcal{R} \left(r x_0 + (1 - r) x_1\right).$$
 * The inverse of $$\mathcal{R}$$ is the set-valued function $$\mathcal{R}^{-1} : Y \rightrightarrows X$$ defined by $$\mathcal{R}^{-1}(y) := \{ x \in X : y \in \mathcal{R}(x) \}.$$ For any subset $$B \subseteq Y,$$ $$\mathcal{R}^{-1}(B) := \cup_{y \in B} \mathcal{R}^{-1}(y).$$
 * 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
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Closed graph theorem
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Uniform boundedness principle
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Open mapping theorem
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Additional corollaries
The following notation and notions are used for these corollaries, where $$\mathcal{R} : X \rightrightarrows Y$$ 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 $\sum_{i=1}^\infty r_i s_i$ where all $$s_i \in S$$ and $\sum_{i=1}^\infty r_i = 1$  is a series of non-negative numbers. If $\sum_{i=1}^\infty r_i s_i$  converges then the series is called convergent while if $$\left(s_i\right)_{i=1}^{\infty}$$ 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 $$X \times Y.$$ Every ideally convex set is lower ideally convex.

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Simons' theorem
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Robinson–Ursescu theorem
The implication (1) $$\implies$$ (2) in the following theorem is known as the Robinson–Ursescu theorem.

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