Ursescu theorem

The following notation and notions are used, where $\backslash mathcal\{R\}\; :\; X\; \backslash rightrightarrows\; Y$ 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 $\backslash operatorname\{aff\}\; S$ and the linear span is denoted by $\backslash operatorname\{span\}\; S.$
• $S^\{i\}\; :=\; \backslash operatorname\{aint\}\_X\; S$ denotes the algebraic interior of $S$ in $X.$
• $\{\}^\{i\}S:=\; \backslash operatorname\{aint\}\_\{\backslash operatorname\{aff\}(S\; -\; S)\}\; S$ denotes the relative algebraic interior of $S$ (i.e. the algebraic interior of $S$ in $\backslash operatorname\{aff\}(S\; -\; S)$).
• $\{\}^\{ib\}S\; :=\; \{\}^\{i\}S$ if $\backslash operatorname\{span\}\; \backslash left(S\; -\; s\_0\backslash right)$ is barreled for some/every $s\_0\; \backslash in\; S$ while $\{\}^\{ib\}S\; :=\; \backslash varnothing$ otherwise.
• If $S$ is convex then it can be shown that for any $x\; \backslash in\; X,$ $x\; \backslash in\; \{\}^\{ib\}\; S$ if and only if the cone generated by $S\; -\; x$ is a barreled linear subspace of $X$ or equivalently, if and only if $\backslash cup\_\{n\; \backslash in\; \backslash N\}\; n\; (S\; -\; x)$ is a barreled linear subspace of $X$
• The domain of $\backslash mathcal\{R\}$ is $\backslash operatorname\{Dom\}\; \backslash mathcal\{R\}\; :=\; \backslash \{\; x\; \backslash in\; X\; :\; \backslash mathcal\{R\}(x)\; \backslash neq\; \backslash varnothing\; \backslash \}.$
• The image of $\backslash mathcal\{R\}$ is $\backslash operatorname\{Im\}\; \backslash mathcal\{R\}\; :=\; \backslash cup\_\{x\; \backslash in\; X\}\; \backslash mathcal\{R\}(x).$ For any subset $A\; \backslash subseteq\; X,$ $\backslash mathcal\{R\}(A)\; :=\; \backslash cup\_\{x\; \backslash in\; A\}\; \backslash mathcal\{R\}(x).$
• The graph of $\backslash mathcal\{R\}$ is $\backslash operatorname\{gr\}\; \backslash mathcal\{R\}\; :=\; \backslash \{\; (x,\; y)\; \backslash in\; X\; \backslash times\; Y\; :\; y\; \backslash in\; \backslash mathcal\{R\}(x)\; \backslash \}.$
• $\backslash mathcal\{R\}$ is closed (respectively, convex) if the graph of $\backslash mathcal\{R\}$ is closed (resp. convex) in $X\; \backslash times\; Y.$
• Note that $\backslash mathcal\{R\}$ is convex if and only if for all $x\_0,\; x\_1\; \backslash in\; X$ and all $r\; \backslash in\; [0,\; 1],$ $r\; \backslash mathcal\{R\}\backslash left(x\_0\backslash right)\; +\; (1\; -\; r)\; \backslash mathcal\{R\}\backslash left(x\_1\backslash right)\; \backslash subseteq\; \backslash mathcal\{R\}\; \backslash left(r\; x\_0\; +\; (1\; -\; r)\; x\_1\backslash right).$
• The inverse of $\backslash mathcal\{R\}$ is the set-valued function $\backslash mathcal\{R\}^\{-1\}\; :\; Y\; \backslash rightrightarrows\; X$ defined by $\backslash mathcal\{R\}^\{-1\}(y)\; :=\; \backslash \{\; x\; \backslash in\; X\; :\; y\; \backslash in\; \backslash mathcal\{R\}(x)\; \backslash \}.$ For any subset $B\; \backslash subseteq\; Y,$ $\backslash mathcal\{R\}^\{-1\}(B)\; :=\; \backslash cup\_\{y\; \backslash in\; B\}\; \backslash mathcal\{R\}^\{-1\}(y).$
• If $f\; :\; X\; \backslash to\; Y$ is a function, then its inverse is the set-valued function $f^\{-1\}\; :\; Y\; \backslash rightrightarrows\; X$ obtained from canonically identifying $f$ with the set-valued function $f\; :\; X\; \backslash rightrightarrows\; Y$ defined by $x\; \backslash mapsto\; \backslash \{\; f(x)\backslash \}.$
• $\backslash operatorname\{int\}\_T\; S$ is the topological interior of $S$ with respect to $T,$ where $S\; \backslash subseteq\; T.$
• $\backslash operatorname\{rint\}\; S\; :=\; \backslash operatorname\{int\}\_\{\backslash operatorname\{aff\}\; S\}\; S$ is the interior of $S$ with respect to $\backslash operatorname\{aff\}\; S.$

{{Math theorem|name=Theorem{{sfn|Zălinescu|2002|p=23}}|note=Ursescu|math_statement=

Let $X$ be a complete semi-metrizable locally convex topological vector space and $\backslash mathcal\{R\}\; :\; X\; \backslash rightrightarrows\; Y$ be a closed convex multifunction with non-empty domain.

Assume that $\backslash operatorname\{span\}\; (\backslash operatorname\{Im\}\; \backslash mathcal\{R\}\; -\; y)$ is a barrelled space for some/every $y\; \backslash in\; \backslash operatorname\{Im\}\; \backslash mathcal\{R\}.$

Assume that $y\_0\; \backslash in\; \{\}^\{i\}(\backslash operatorname\{Im\}\; \backslash mathcal\{R\})$ and let $x\_0\; \backslash in\; \backslash mathcal\{R\}^\{-1\}\backslash left(y\_0\backslash right)$ (so that $y\_0\; \backslash in\; \backslash mathcal\{R\}\backslash left(x\_0\backslash right)$).

Then for every neighborhood $U$ of $x\_0$ in $X,$ $y\_0$ belongs to the relative interior of $\backslash mathcal\{R\}(U)$ in $\backslash operatorname\{aff\}\; (\backslash operatorname\{Im\}\; \backslash mathcal\{R\})$ (that is, $y\_0\; \backslash in\; \backslash operatorname\{int\}\_\{\backslash operatorname\{aff\}\; (\backslash operatorname\{Im\}\; \backslash mathcal\{R\})\}\; \backslash mathcal\{R\}(U)$).

In particular, if $\{\}^\{ib\}(\backslash operatorname\{Im\}\; \backslash mathcal\{R\})\; \backslash neq\; \backslash varnothing$ then $\{\}^\{ib\}(\backslash operatorname\{Im\}\; \backslash mathcal\{R\})\; =\; \{\}^\{i\}(\backslash operatorname\{Im\}\; \backslash mathcal\{R\})\; =\; \backslash operatorname\{rint\}\; (\backslash operatorname\{Im\}\; \backslash mathcal\{R\}).$

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{{Math theorem|name=Closed graph theorem |math_statement=

Let $X$ and $Y$ be Fréchet spaces and $T\; :\; X\; \backslash to\; Y$ be a linear map. Then $T$ is continuous if and only if the graph of $T$ is closed in $X\; \backslash times\; Y.$

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{{math proof|

For the non-trivial direction, assume that the graph of $T$ is closed and let $\backslash mathcal\{R\}\; :=\; T^\{-1\}\; :\; Y\; \backslash rightrightarrows\; X.$ It is easy to see that $\backslash operatorname\{gr\}\; \backslash mathcal\{R\}$ is closed and convex and that its image is $X.$

Given $x\; \backslash in\; X,$ $(Tx,\; x)$ belongs to $Y\; \backslash times\; X$ so that for every open neighborhood $V$ of $Tx$ in $Y,$ $\backslash mathcal\{R\}(V)\; =\; T^\{-1\}(V)$ is a neighborhood of $x$ in $X.$

Thus $T$ is continuous at $x.$

Q.E.D.

}}

{{Math theorem|name=Uniform boundedness principle|math_statement=

Let $X$ and $Y$ be Fréchet spaces and $T\; :\; X\; \backslash to\; Y$ be a bijective linear map. Then $T$ is continuous if and only if $T^\{-1\}\; :\; Y\; \backslash to\; X$ is continuous. Furthermore, if $T$ is continuous then $T$ is an isomorphism of Fréchet spaces.

}}

{{math proof|Apply the closed graph theorem to $T$ and $T^\{-1\}.$

Q.E.D.}}

{{Math theorem|name=Open mapping theorem |math_statement=

Let $X$ and $Y$ be Fréchet spaces and $T\; :\; X\; \backslash to\; Y$ be a continuous surjective linear map. Then T is an open map.

}}

{{math proof|

Clearly, $T$ is a closed and convex relation whose image is $Y.$

Let $U$ be a non-empty open subset of $X,$ let $y$ be in $T(U),$ and let $x$ in $U$ be such that $y\; =\; Tx.$

From the Ursescu theorem it follows that $T(U)$ is a neighborhood of $y.$ Q.E.D.}}

The following notation and notions are used for these corollaries, where $\backslash mathcal\{R\}\; :\; X\; \backslash rightrightarrows\; Y$ 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 $\backslash sum\_\{i=1\}^\backslash infty\; r\_i\; s\_i$ where all $s\_i\; \backslash in\; S$ and $\backslash sum\_\{i=1\}^\backslash infty\; r\_i\; =\; 1$ is a series of non-negative numbers. If $\backslash sum\_\{i=1\}^\backslash infty\; r\_i\; s\_i$ converges then the series is called convergent while if $\backslash left(s\_i\backslash right)\_\{i=1\}^\{\backslash infty\}$ 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 $X\; \backslash times\; Y.$ Every ideally convex set is lower ideally convex.

{{Math theorem |name=Corollary |math_statement=

Let $X$ be a barreled first countable space and let $C$ be a subset of $X.$ Then:

1. If $C$ is lower ideally convex then $C^\{i\}\; =\; \backslash operatorname\{int\}\; C.$
2. If $C$ is ideally convex then $C^\{i\}\; =\; \backslash operatorname\{int\}\; C\; =\; \backslash operatorname\{int\}\; \backslash left(\backslash operatorname\{cl\}\; C\backslash right)\; =\; \backslash left(\backslash operatorname\{cl\}\; C\backslash right)^i.$

}}

{{Math theorem|name=Simons' theorem{{sfn|Zălinescu|2002|p=22-23}}|note=|math_statement=

Let $X$ and $Y$ be first countable with $X$ locally convex. Suppose that $\backslash mathcal\{R\}\; :\; X\; \backslash rightrightarrows\; Y$ is a multimap with non-empty domain that satisfies condition (Hwx) or else assume that $X$ is a Fréchet space and that $\backslash mathcal\{R\}$ is lower ideally convex.

Assume that $\backslash operatorname\{span\}\; (\backslash operatorname\{Im\}\; \backslash mathcal\{R\}\; -\; y)$ is barreled for some/every $y\; \backslash in\; \backslash operatorname\{Im\}\; \backslash mathcal\{R\}.$

Assume that $y\_0\; \backslash in\; \{\}^\{i\}(\backslash operatorname\{Im\}\; \backslash mathcal\{R\})$ and let $x\_0\; \backslash in\; \backslash mathcal\{R\}^\{-1\}\backslash left(y\_0\backslash right).$

Then for every neighborhood $U$ of $x\_0$ in $X,$ $y\_0$ belongs to the relative interior of $\backslash mathcal\{R\}(U)$ in $\backslash operatorname\{aff\}\; (\backslash operatorname\{Im\}\; \backslash mathcal\{R\})$ (i.e. $y\_0\; \backslash in\; \backslash operatorname\{int\}\_\{\backslash operatorname\{aff\}\; (\backslash operatorname\{Im\}\; \backslash mathcal\{R\})\}\; \backslash mathcal\{R\}(U)$).

In particular, if $\{\}^\{ib\}(\backslash operatorname\{Im\}\; \backslash mathcal\{R\})\; \backslash neq\; \backslash varnothing$ then $\{\}^\{ib\}(\backslash operatorname\{Im\}\; \backslash mathcal\{R\})\; =\; \{\}^\{i\}(\backslash operatorname\{Im\}\; \backslash mathcal\{R\})\; =\; \backslash operatorname\{rint\}\; (\backslash operatorname\{Im\}\; \backslash mathcal\{R\}).$

}}

The implication (1) $\backslash implies$ (2) in the following theorem is known as the Robinson–Ursescu theorem.{{sfn|Zălinescu|2002|p=24}}

{{Math theorem|name=Robinson–Ursescu theorem{{sfn|Zălinescu|2002|p=24}} |note=|math_statement=

Let $(X,\; \backslash |\backslash ,\backslash cdot\backslash ,\backslash |)$ and $(Y,\; \backslash |\backslash ,\backslash cdot\backslash ,\backslash |)$ be normed spaces and $\backslash mathcal\{R\}\; :\; X\; \backslash rightrightarrows\; Y$ be a multimap with non-empty domain.

Suppose that $Y$ is a barreled space, the graph of $\backslash mathcal\{R\}$ verifies condition condition (Hwx), and that $(x\_0,\; y\_0)\; \backslash in\; \backslash operatorname\{gr\}\; \backslash mathcal\{R\}.$

Let $C\_X$ (resp. $C\_Y$) denote the closed unit ball in $X$ (resp. $Y$) (so $C\_X\; =\; \backslash \{\; x\; \backslash in\; X\; :\; \backslash |\; x\; \backslash |\; \backslash leq\; 1\; \backslash \}$).

Then the following are equivalent:

1. $y\_0$ belongs to the algebraic interior of $\backslash operatorname\{Im\}\; \backslash mathcal\{R\}.$
2. $y\_0\; \backslash in\; \backslash operatorname\{int\}\; \backslash mathcal\{R\}\backslash left(x\_0\; +\; C\_X\backslash right).$
3. There exists $B\; >\; 0$ such that for all $0\; \backslash leq\; r\; \backslash leq\; 1,$ $y\_0\; +\; B\; r\; C\_Y\; \backslash subseteq\; \backslash mathcal\{R\}\; \backslash left(x\_0\; +\; r\; C\_X\backslash right).$
4. There exist $A\; >\; 0$ and $B\; >\; 0$ such that for all $x\; \backslash in\; x\_0\; +\; A\; C\_X$ and all $y\; \backslash in\; y\_0\; +\; A\; C\_Y,$ $d\backslash left(x,\; \backslash mathcal\{R\}^\{-1\}(y)\backslash right)\; \backslash leq\; B\; \backslash cdot\; d(y,\; \backslash mathcal\{R\}(x)).$
5. There exists $B\; >\; 0$ such that for all $x\; \backslash in\; X$ and all $y\; \backslash in\; y\_0\; +\; B\; C\_Y,$ $d\; \backslash left(x,\; \backslash mathcal\{R\}^\{-1\}(y)\backslash right)\; \backslash leq\; \backslash frac\{1\; +\; \backslash left\backslash |x\; -\; x\_0\backslash right\backslash |\}\{B\; -\; \backslash left\backslash |y\; -\; y\_0\backslash right\backslash |\}\; \backslash cdot\; d(y,\; \backslash mathcal\{R\}(x)).$

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• {{annotated link|Closed graph theorem}}
• {{annotated link|Closed graph theorem (functional analysis)}}
• {{annotated link|Open mapping theorem (functional analysis)}}
• {{annotated link|Surjection of Fréchet spaces}}
• {{annotated link|Uniform boundedness principle}}
• {{annotated link|Webbed space}}

{{reflist|group=note}}

{{reflist}}

• {{Zălinescu Convex Analysis in General Vector Spaces 2002}}
• {{Cite journal|last=Baggs|first=Ivan|date=1974|title=Functions with a closed graph|url=https://www.ams.org/|journal=Proceedings of the American Mathematical Society|language=en|volume=43|issue=2|pages=439–442|doi=10.1090/S0002-9939-1974-0334132-8|issn=0002-9939|doi-access=free}}

{{Convex analysis and variational analysis}}

{{Functional analysis}}

{{Topological vector spaces}}

Category:Theorems involving convexity

Category:Theorems in functional analysis