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 ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}$ is a multifunction and S is a non-empty subset of a topological vector space X:

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

Statement

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

Corollaries

Closed graph theorem

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

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

Uniform boundedness principle

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

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

Open mapping theorem

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

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

Additional corollaries

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

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

Related theorems

Simons' theorem

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

Robinson–Ursescu theorem

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

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

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

See also

• Closed graph theorem
• Uniform boundedness principle

Notes

1. Zalinescu 2002, p. 23.
2. Zalinescu 2002, p. 22-23.
3. Zalinescu 2002, p. 24.

References

• Zalinescu, C (2002). Convex analysis in general vector spaces. River Edge, N.J. London: World Scientific. ISBN 981-238-067-1. OCLC 285163112.CS1 maint: ref=harv (link)
• 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.