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 set-valued function and ${\displaystyle S}$ is a non-empty subset of a topological vector space ${\displaystyle X}$:

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

Statement

Theorem ^[1]  (Ursescu) — Let ${\displaystyle X}$ 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 ${\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)}$ is a barrelled space for some/every ${\displaystyle y\in \operatorname {Im} {\mathcal {R}}.}$ Assume that ${\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})}$ 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 ${\displaystyle U}$ of ${\displaystyle x_{0}}$ in ${\displaystyle X,}$${\displaystyle y_{0}}$ belongs to the relative interior of ${\displaystyle {\mathcal {R}}(U)}$ in ${\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})}$ (that is, ${\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}$). In particular, if ${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing }$ then ${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}$

Corollaries

Closed graph theorem

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

Proof

For the non-trivial direction, assume that the graph of ${\displaystyle T}$ 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 ${\displaystyle X.}$ Given ${\displaystyle x\in X,}$${\displaystyle (Tx,x)}$ belongs to ${\displaystyle Y\times X}$ so that for every open neighborhood ${\displaystyle V}$ of ${\displaystyle Tx}$ in ${\displaystyle Y,}$${\displaystyle {\mathcal {R}}(V)=T^{-1}(V)}$ is a neighborhood of ${\displaystyle x}$ in ${\displaystyle X.}$ Thus ${\displaystyle T}$ is continuous at ${\displaystyle x.}$ Q.E.D.

Uniform boundedness principle

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

Proof

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

Open mapping theorem

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

Proof

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

Additional corollaries

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

• a convex series with elements of ${\displaystyle S}$ is a series of the form ${\textstyle \sum _{i=1}^{\infty }r_{i}s_{i}}$ where all ${\displaystyle s_{i}\in S}$ 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 called convergent while if ${\displaystyle \left(s_{i}\right)_{i=1}^{\infty }}$ is bounded then the series is called bounded and b-convex.
• ${\displaystyle S}$ is ideally convex if any convergent b-convex series of elements of ${\displaystyle S}$ has its sum in ${\displaystyle S.}$
• ${\displaystyle S}$ is lower ideally convex if there exists a Fréchet space ${\displaystyle Y}$ such that ${\displaystyle S}$ is equal to the projection onto ${\displaystyle X}$ of some ideally convex subset B of ${\displaystyle X\times Y.}$ Every ideally convex set is lower ideally convex.

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

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

Related theorems

Simons' theorem

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

Robinson–Ursescu theorem

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

Robinson–Ursescu theorem ^[3]  — Let ${\displaystyle (X,\|\,\cdot \,\|)}$ and ${\displaystyle (Y,\|\,\cdot \,\|)}$ be normed spaces and ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}$ be a multimap with non-empty domain. Suppose that ${\displaystyle Y}$ 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 ${\displaystyle X}$ (resp. ${\displaystyle Y}$) (so ${\displaystyle C_{X}=\{x\in X:\|x\|\leq 1\}}$). 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(y,{\mathcal {R}}(x)).}$
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(y,{\mathcal {R}}(x)).}$

See also

• Closed graph theorem  – Theorem relating continuity to graphs
• Closed graph theorem (functional analysis)  – Theorems connecting continuity to closure of graphs
• Open mapping theorem (functional analysis)  – Condition for a linear operator to be open
• Surjection of Fréchet spaces  – Characterization of surjectivity
• Uniform boundedness principle  – A theorem stating that pointwise boundedness implies uniform boundedness
• Webbed space  – Space where open mapping and closed graph theorems hold

Notes

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

References

• 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 – via Internet 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.