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Ursescu theorem

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Generalization of closed graph, open mapping, and uniform boundedness theorem

In mathematics, particularly infunctional analysis andconvex analysis, theUrsescu theorem is a theorem that generalizes theclosed graph theorem, theopen mapping theorem, and theuniform boundedness principle.

Ursescu theorem

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The following notation and notions are used, where ${\mathcal {R}}:X\rightrightarrows Y$ is aset-valued function and $S {\displaystyle S}$ is a non-empty subset of atopological vector space $X {\displaystyle X}$ :

theaffine span of $S {\displaystyle S}$ is denoted by $\operatorname {aff} S$ and thelinear span is denoted by $\operatorname {span} S.$
$S^{i}:=\operatorname {aint} _{X}S$ denotes thealgebraic interior of $S {\displaystyle S}$ in $X . {\displaystyle X.}$
${}^{i}S:=\operatorname {aint} _{\operatorname {aff} (S-S)}S$ denotes therelative algebraic interior of $S {\displaystyle S}$ (i.e. the algebraic interior of $S {\displaystyle S}$ in $\operatorname {aff} (S-S)$ ).
${}^{ib}S:={}^{i}S$ if $\operatorname {span} \left(S-s_{0}\right)$ isbarreled for some/every $s_{0}\in S$ while ${}^{ib}S:=\varnothing$ otherwise.
- If $S {\displaystyle S}$ 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 $X {\displaystyle X}$ or equivalently, if and only if $\cup _{n\in \mathbb {N} }n(S-x)$ is a barreled linear subspace of $X {\displaystyle X}$
Thedomain of ${\mathcal {R}}$ is $\operatorname {Dom} {\mathcal {R}}:=\{x\in X:{\mathcal {R}}(x)\neq \varnothing \}.$
Theimage of ${\mathcal {R}}$ is $\operatorname {Im} {\mathcal {R}}:=\cup _{x\in X}{\mathcal {R}}(x).$ For any subset $A\subseteq X,$ ${\mathcal {R}}(A):=\cup _{x\in A}{\mathcal {R}}(x).$
Thegraph of ${\mathcal {R}}$ is $\operatorname {gr} {\mathcal {R}}:=\{(x,y)\in X\times Y:y\in {\mathcal {R}}(x)\}.$
${\mathcal {R}}$ isclosed (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 $r\in [0,1],$ $r{\mathcal {R}}\left(x_{0}\right)+(1-r){\mathcal {R}}\left(x_{1}\right)\subseteq {\mathcal {R}}\left(rx_{0}+(1-r)x_{1}\right).$
Theinverse 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 $f {\displaystyle f}$ with the set-valued function $f:X\rightrightarrows Y$ defined by $x\mapsto \{f(x)\}.$
$\operatorname {int} _{T}S$ is thetopological interior of $S {\displaystyle S}$ with respect to $T, {\displaystyle T,}$ where $S\subseteq T.$
$\operatorname {rint} S:=\operatorname {int} _{\operatorname {aff} S}S$ is theinterior of $S {\displaystyle S}$ with respect to $\operatorname {aff} S.$

Statement

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Corollaries

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Closed graph theorem

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Closed graph theorem—Let $X {\displaystyle X}$ and $Y {\displaystyle Y}$ beFréchet spaces and $T:X\to Y$ be a linear map. Then $T {\displaystyle T}$ is continuous if and only if the graph of $T {\displaystyle T}$ is closed in $X\times Y.$

Proof

For the non-trivial direction, assume that the graph of $T {\displaystyle T}$ is closed and let ${\mathcal {R}}:=T^{-1}:Y\rightrightarrows X.$ It is easy to see that $\operatorname {gr} {\mathcal {R}}$ is closed and convex and that its image is $X . {\displaystyle X.}$ Given $x\in X,$ $(Tx,x)$ belongs to $Y\times X$ so that for every open neighborhood $V {\displaystyle V}$ of $T x {\displaystyle Tx}$ in $Y, {\displaystyle Y,}$ ${\mathcal {R}}(V)=T^{-1}(V)$ is a neighborhood of $x {\displaystyle x}$ in $X . {\displaystyle X.}$ Thus $T {\displaystyle T}$ is continuous at $x . {\displaystyle x.}$ Q.E.D.

Uniform boundedness principle

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Uniform boundedness principle—Let $X {\displaystyle X}$ and $Y {\displaystyle Y}$ beFréchet spaces and $T:X\to Y$ be a bijective linear map. Then $T {\displaystyle T}$ is continuous if and only if $T^{-1}:Y\to X$ is continuous. Furthermore, if $T {\displaystyle T}$ is continuous then $T {\displaystyle T}$ is an isomorphism ofFréchet spaces.

Proof

Apply the closed graph theorem to $T {\displaystyle T}$ and $T^{-1}.$ Q.E.D.

Open mapping theorem

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Open mapping theorem—Let $X {\displaystyle X}$ and $Y {\displaystyle Y}$ beFréchet spaces and $T:X\to Y$ be a continuous surjective linear map. Then T is anopen map.

Proof

Clearly, $T {\displaystyle T}$ is a closed and convex relation whose image is $Y . {\displaystyle Y.}$ Let $U {\displaystyle U}$ be a non-empty open subset of $X, {\displaystyle X,}$ let $y {\displaystyle y}$ be in $T(U),$ and let $x {\displaystyle x}$ in $U {\displaystyle U}$ be such that $y=Tx.$ From the Ursescu theorem it follows that $T(U)$ is a neighborhood of $y . {\displaystyle y.}$ Q.E.D.

Additional corollaries

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The following notation and notions are used for these corollaries, where ${\mathcal {R}}:X\rightrightarrows Y$ is a set-valued function, $S {\displaystyle S}$ is a non-empty subset of atopological vector space $X {\displaystyle X}$ :

aconvex series with elements of $S {\displaystyle S}$ is aseries of the form ${\textstyle \sum _{i=1}^{\infty }r_{i}s_{i}}$ where all $s_{i}\in S$ and ${\textstyle \sum _{i=1}^{\infty }r_{i}=1}$ is a series of non-negative numbers. If ${\textstyle \sum _{i=1}^{\infty }r_{i}s_{i}}$ converges then the series is calledconvergent while if $\left(s_{i}\right)_{i=1}^{\infty }$ is bounded then the series is calledbounded andb-convex.
$S {\displaystyle S}$ isideally convex if any convergent b-convex series of elements of $S {\displaystyle S}$ has its sum in $S . {\displaystyle S.}$
$S {\displaystyle S}$ islower ideally convex if there exists aFréchet space $Y {\displaystyle Y}$ such that $S {\displaystyle S}$ is equal to the projection onto $X {\displaystyle X}$ of some ideally convex subsetB of $X\times Y.$ Every ideally convex set is lower ideally convex.

Corollary—Let $X {\displaystyle X}$ be a barreledfirst countable space and let $C {\displaystyle C}$ be a subset of $X . {\displaystyle X.}$ Then:

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

Related theorems

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Simons' theorem

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Robinson–Ursescu theorem

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The implication (1) $\implies$ (2) in the following theorem is known as the Robinson–Ursescu theorem.^[3]

Robinson–Ursescu theorem^[3]—Let $(X,\|\,\cdot \,\|)$ and $(Y,\|\,\cdot \,\|)$ benormed spaces and ${\mathcal {R}}:X\rightrightarrows Y$ be a multimap with non-empty domain. Suppose that $Y {\displaystyle Y}$ is abarreled space, the graph of ${\mathcal {R}}$ verifies conditioncondition (Hwx), and that $(x_{0},y_{0})\in \operatorname {gr} {\mathcal {R}}.$ Let $C_{X}$ (resp. $C_{Y}$ ) denote the closed unit ball in $X {\displaystyle X}$ (resp. $Y {\displaystyle Y}$ ) (so $C_{X}=\{x\in X:\|x\|\leq 1\}$ ). Then the following are equivalent:

$y_{0}$ belongs to thealgebraic interior of $\operatorname {Im} {\mathcal {R}}.$
$y_{0}\in \operatorname {int} {\mathcal {R}}\left(x_{0}+C_{X}\right).$
There exists $B>0$ such that for all $0\leq r\leq 1,$ $y_{0}+BrC_{Y}\subseteq {\mathcal {R}}\left(x_{0}+rC_{X}\right).$
There exist $A>0$ and $B>0$ such that for all $x\in x_{0}+AC_{X}$ and all $y\in y_{0}+AC_{Y},$ $d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq B\cdot d(y,{\mathcal {R}}(x)).$
There exists $B>0$ such that for all $x\in X$ and all $y\in y_{0}+BC_{Y},$ $d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq {\frac {1+\left\|x-x_{0}\right\|}{B-\left\|y-y_{0}\right\|}}\cdot d(y,{\mathcal {R}}(x)).$

Notes

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References

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Zălinescu, Constantin (30 July 2002).Convex Analysis in General Vector Spaces. River Edge, N.J. London:World Scientific Publishing.ISBN 978-981-4488-15-0.MR 1921556.OCLC 285163112 – viaInternet Archive.
Baggs, Ivan (1974)."Functions with a closed graph".Proceedings of the American Mathematical Society.43 (2):439–442.doi:10.1090/S0002-9939-1974-0334132-8.ISSN 0002-9939.

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