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 $\mathcal : X \rightrightarrows Y$ 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 $\operatorname S$ and the linear span is denoted by $\operatorname S.$
* $S^ := \operatorname_X S$ denotes the algebraic interior of $S$ in $X.$
* $^S:= \operatorname_ S$ denotes the relative algebraic interior of $S$ (i.e. the algebraic interior of $S$ in $\operatorname(S - S)$).
* $^S := ^S$ if $\operatorname \left(S - s_0\right)$ is barreled for some/every $s_0 \in S$ while $^S := \varnothing$ otherwise.
** If $S$ is convex then it can be shown that for any $x \in X,$ $x \in ^ S$ if and only if the cone generated by $S - x$ is a barreled linear subspace of $X$ or equivalently, if and only if $\cup_ n (S - x)$ is a barreled linear subspace of $X$
* The domain of $\mathcal$ is $\operatorname \mathcal := \.$
* The image of $\mathcal$ is $\operatorname \mathcal := \cup_ \mathcal(x).$ For any subset $A \subseteq X,$ $\mathcal(A) := \cup_ \mathcal(x).$
* The graph of $\mathcal$ is $\operatorname \mathcal := \.$
* $\mathcal$ is closed (respectively, convex) if the graph of $\mathcal$ is closed (resp. convex) in $X \times Y.$
** Note that $\mathcal$ is convex if and only if for all $x_0, x_1 \in X$ and all $r \in$, 1 $r \mathcal\left(x_0\right) + (1 - r) \mathcal\left(x_1\right) \subseteq \mathcal \left(r x_0 + (1 - r) x_1\right).$
* The inverse of $\mathcal$ is the set-valued function $\mathcal^ : Y \rightrightarrows X$ defined by $\mathcal^(y) := \.$ For any subset $B \subseteq Y,$ $\mathcal^(B) := \cup_ \mathcal^(y).$
** If $f : X \to Y$ is a function, then its inverse is the set-valued function $f^ : Y \rightrightarrows X$ obtained from canonically identifying $f$ with the set-valued function $f : X \rightrightarrows Y$ defined by $x \mapsto \.$
* $\operatorname_T S$ is the topological interior of $S$ with respect to $T,$ where $S \subseteq T.$
* $\operatorname S := \operatorname_ S$ is the interior of $S$ with respect to $\operatorname S.$

Statement

Corollaries

Closed graph theorem

Uniform boundedness principle

Open mapping theorem

Additional corollaries

The following notation and notions are used for these corollaries, where $\mathcal : X \rightrightarrows Y$ 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 $\sum_^\infty r_i s_i$ where all $s_i \in S$ and $\sum_^\infty r_i = 1$ is a series of non-negative numbers. If $\sum_^\infty r_i s_i$ converges then the series is called convergent while if $\left(s_i\right)_^$ 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.

Related theorems

Simons' theorem

Robinson–Ursescu theorem

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

See also

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Notes

References

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{{Topological vector spaces

Theorems involving convexity
Theorems in functional analysis