Ursescu theorem

Ursescu theorem

Statement

Theorem^[1]  (Ursescu)—Let ${\displaystyle X}${\displaystyle X} be a complete semi-metrizable locally convex topological vector space and ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}${\displaystyle {\mathcal {R}}:X\rightrightarrows Y} be a closed convex multifunction with non-empty domain. Assume that ${\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)}${\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)} is a barrelled space for some/every ${\displaystyle y\in \operatorname {Im} {\mathcal {R}}.}${\displaystyle y\in \operatorname {Im} {\mathcal {R}}.} Assume that ${\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})}${\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})} and let ${\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right)}${\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right)} (so that ${\displaystyle y_{0}\in {\mathcal {R}}\left(x_{0}\right)}${\displaystyle y_{0}\in {\mathcal {R}}\left(x_{0}\right)}). Then for every neighborhood ${\displaystyle U}${\displaystyle U} of ${\displaystyle x_{0}}${\displaystyle x_{0}} in ${\displaystyle X,}${\displaystyle X,} ${\displaystyle y_{0}}${\displaystyle y_{0}} belongs to the relative interior of ${\displaystyle {\mathcal {R}}(U)}${\displaystyle {\mathcal {R}}(U)} in ${\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})}${\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})} (that is, ${\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}${\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}). In particular, if ${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing }${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing } then ${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}

Corollaries

Closed graph theorem

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

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

Uniform boundedness principle

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

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

Open mapping theorem

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

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

Additional corollaries

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

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

Related theorems

Simons' theorem

Simons' theorem^[2] —Let ${\displaystyle X}${\displaystyle X} and ${\displaystyle Y}${\displaystyle Y} be first countable with ${\displaystyle X}${\displaystyle X} locally convex. Suppose that ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}${\displaystyle {\mathcal {R}}:X\rightrightarrows Y} is a multimap with non-empty domain that satisfies condition (Hwx) or else assume that ${\displaystyle X}${\displaystyle X} is a Fréchet space and that ${\displaystyle {\mathcal {R}}}${\displaystyle {\mathcal {R}}} is lower ideally convex. Assume that ${\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)}${\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)} is barreled for some/every ${\displaystyle y\in \operatorname {Im} {\mathcal {R}}.}${\displaystyle y\in \operatorname {Im} {\mathcal {R}}.} Assume that ${\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})}${\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})} and let ${\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right).}${\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right).} Then for every neighborhood ${\displaystyle U}${\displaystyle U} of ${\displaystyle x_{0}}${\displaystyle x_{0}} in ${\displaystyle X,}${\displaystyle X,} ${\displaystyle y_{0}}${\displaystyle y_{0}} belongs to the relative interior of ${\displaystyle {\mathcal {R}}(U)}${\displaystyle {\mathcal {R}}(U)} in ${\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})}${\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})} (i.e. ${\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}${\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}). In particular, if ${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing }${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing } then ${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}

Robinson–Ursescu theorem

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

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

See also

Notes

References