Ursescu theorem

Short description: Generalization of closed graph, open mapping, and uniform boundedness 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 [math]\displaystyle{ \mathcal{R} : X \rightrightarrows Y }[/math] is a set-valued function and [math]\displaystyle{ S }[/math] is a non-empty subset of a topological vector space [math]\displaystyle{ X }[/math]:

the affine span of [math]\displaystyle{ S }[/math] is denoted by [math]\displaystyle{ \operatorname{aff} S }[/math] and the linear span is denoted by [math]\displaystyle{ \operatorname{span} S. }[/math]
[math]\displaystyle{ S^{i} := \operatorname{aint}_X S }[/math] denotes the algebraic interior of [math]\displaystyle{ S }[/math] in [math]\displaystyle{ X. }[/math]
[math]\displaystyle{ {}^{i}S:= \operatorname{aint}_{\operatorname{aff}(S - S)} S }[/math] denotes the relative algebraic interior of [math]\displaystyle{ S }[/math] (i.e. the algebraic interior of [math]\displaystyle{ S }[/math] in [math]\displaystyle{ \operatorname{aff}(S - S) }[/math]).
[math]\displaystyle{ {}^{ib}S := {}^{i}S }[/math] if [math]\displaystyle{ \operatorname{span} \left(S - s_0\right) }[/math] is barreled for some/every [math]\displaystyle{ s_0 \in S }[/math] while [math]\displaystyle{ {}^{ib}S := \varnothing }[/math] otherwise.
- If [math]\displaystyle{ S }[/math] is convex then it can be shown that for any [math]\displaystyle{ x \in X, }[/math] [math]\displaystyle{ x \in {}^{ib} S }[/math] if and only if the cone generated by [math]\displaystyle{ S - x }[/math] is a barreled linear subspace of [math]\displaystyle{ X }[/math] or equivalently, if and only if [math]\displaystyle{ \cup_{n \in \N} n (S - x) }[/math] is a barreled linear subspace of [math]\displaystyle{ X }[/math]
The domain of [math]\displaystyle{ \mathcal{R} }[/math] is [math]\displaystyle{ \operatorname{Dom} \mathcal{R} := \{ x \in X : \mathcal{R}(x) \neq \varnothing \}. }[/math]
The image of [math]\displaystyle{ \mathcal{R} }[/math] is [math]\displaystyle{ \operatorname{Im} \mathcal{R} := \cup_{x \in X} \mathcal{R}(x). }[/math] For any subset [math]\displaystyle{ A \subseteq X, }[/math] [math]\displaystyle{ \mathcal{R}(A) := \cup_{x \in A} \mathcal{R}(x). }[/math]
The graph of [math]\displaystyle{ \mathcal{R} }[/math] is [math]\displaystyle{ \operatorname{gr} \mathcal{R} := \{ (x, y) \in X \times Y : y \in \mathcal{R}(x) \}. }[/math]
[math]\displaystyle{ \mathcal{R} }[/math] is closed (respectively, convex) if the graph of [math]\displaystyle{ \mathcal{R} }[/math] is closed (resp. convex) in [math]\displaystyle{ X \times Y. }[/math]
- Note that [math]\displaystyle{ \mathcal{R} }[/math] is convex if and only if for all [math]\displaystyle{ x_0, x_1 \in X }[/math] and all [math]\displaystyle{ r \in [0, 1], }[/math] [math]\displaystyle{ r \mathcal{R}\left(x_0\right) + (1 - r) \mathcal{R}\left(x_1\right) \subseteq \mathcal{R} \left(r x_0 + (1 - r) x_1\right). }[/math]
The inverse of [math]\displaystyle{ \mathcal{R} }[/math] is the set-valued function [math]\displaystyle{ \mathcal{R}^{-1} : Y \rightrightarrows X }[/math] defined by [math]\displaystyle{ \mathcal{R}^{-1}(y) := \{ x \in X : y \in \mathcal{R}(x) \}. }[/math] For any subset [math]\displaystyle{ B \subseteq Y, }[/math] [math]\displaystyle{ \mathcal{R}^{-1}(B) := \cup_{y \in B} \mathcal{R}^{-1}(y). }[/math]
- If [math]\displaystyle{ f : X \to Y }[/math] is a function, then its inverse is the set-valued function [math]\displaystyle{ f^{-1} : Y \rightrightarrows X }[/math] obtained from canonically identifying [math]\displaystyle{ f }[/math] with the set-valued function [math]\displaystyle{ f : X \rightrightarrows Y }[/math] defined by [math]\displaystyle{ x \mapsto \{ f(x)\}. }[/math]
[math]\displaystyle{ \operatorname{int}_T S }[/math] is the topological interior of [math]\displaystyle{ S }[/math] with respect to [math]\displaystyle{ T, }[/math] where [math]\displaystyle{ S \subseteq T. }[/math]
[math]\displaystyle{ \operatorname{rint} S := \operatorname{int}_{\operatorname{aff} S} S }[/math] is the interior of [math]\displaystyle{ S }[/math] with respect to [math]\displaystyle{ \operatorname{aff} S. }[/math]

Statement

Theorem^[1] (Ursescu) — Let [math]\displaystyle{ X }[/math] be a complete semi-metrizable locally convex topological vector space and [math]\displaystyle{ \mathcal{R} : X \rightrightarrows Y }[/math] be a closed convex multifunction with non-empty domain. Assume that [math]\displaystyle{ \operatorname{span} (\operatorname{Im} \mathcal{R} - y) }[/math] is a barrelled space for some/every [math]\displaystyle{ y \in \operatorname{Im} \mathcal{R}. }[/math] Assume that [math]\displaystyle{ y_0 \in {}^{i}(\operatorname{Im} \mathcal{R}) }[/math] and let [math]\displaystyle{ x_0 \in \mathcal{R}^{-1}\left(y_0\right) }[/math] (so that [math]\displaystyle{ y_0 \in \mathcal{R}\left(x_0\right) }[/math]). Then for every neighborhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ x_0 }[/math] in [math]\displaystyle{ X, }[/math] [math]\displaystyle{ y_0 }[/math] belongs to the relative interior of [math]\displaystyle{ \mathcal{R}(U) }[/math] in [math]\displaystyle{ \operatorname{aff} (\operatorname{Im} \mathcal{R}) }[/math] (that is, [math]\displaystyle{ y_0 \in \operatorname{int}_{\operatorname{aff} (\operatorname{Im} \mathcal{R})} \mathcal{R}(U) }[/math]). In particular, if [math]\displaystyle{ {}^{ib}(\operatorname{Im} \mathcal{R}) \neq \varnothing }[/math] then [math]\displaystyle{ {}^{ib}(\operatorname{Im} \mathcal{R}) = {}^{i}(\operatorname{Im} \mathcal{R}) = \operatorname{rint} (\operatorname{Im} \mathcal{R}). }[/math]

Corollaries

Closed graph theorem

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

Proof

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

Uniform boundedness principle

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

Proof

Apply the closed graph theorem to [math]\displaystyle{ T }[/math] and [math]\displaystyle{ T^{-1}. }[/math] Q.E.D.

Open mapping theorem

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

Proof

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

Additional corollaries

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

a convex series with elements of [math]\displaystyle{ S }[/math] is a series of the form [math]\displaystyle{ \sum_{i=1}^\infty r_i s_i }[/math] where all [math]\displaystyle{ s_i \in S }[/math] and [math]\displaystyle{ \sum_{i=1}^\infty r_i = 1 }[/math] is a series of non-negative numbers. If [math]\displaystyle{ \sum_{i=1}^\infty r_i s_i }[/math] converges then the series is called convergent while if [math]\displaystyle{ \left(s_i\right)_{i=1}^{\infty} }[/math] is bounded then the series is called bounded and b-convex.
[math]\displaystyle{ S }[/math] is ideally convex if any convergent b-convex series of elements of [math]\displaystyle{ S }[/math] has its sum in [math]\displaystyle{ S. }[/math]
[math]\displaystyle{ S }[/math] is lower ideally convex if there exists a Fréchet space [math]\displaystyle{ Y }[/math] such that [math]\displaystyle{ S }[/math] is equal to the projection onto [math]\displaystyle{ X }[/math] of some ideally convex subset B of [math]\displaystyle{ X \times Y. }[/math] Every ideally convex set is lower ideally convex.

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

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

Related theorems

Simons' theorem

Simons' theorem^[2] — Let [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] be first countable with [math]\displaystyle{ X }[/math] locally convex. Suppose that [math]\displaystyle{ \mathcal{R} : X \rightrightarrows Y }[/math] is a multimap with non-empty domain that satisfies condition (Hwx) or else assume that [math]\displaystyle{ X }[/math] is a Fréchet space and that [math]\displaystyle{ \mathcal{R} }[/math] is lower ideally convex. Assume that [math]\displaystyle{ \operatorname{span} (\operatorname{Im} \mathcal{R} - y) }[/math] is barreled for some/every [math]\displaystyle{ y \in \operatorname{Im} \mathcal{R}. }[/math] Assume that [math]\displaystyle{ y_0 \in {}^{i}(\operatorname{Im} \mathcal{R}) }[/math] and let [math]\displaystyle{ x_0 \in \mathcal{R}^{-1}\left(y_0\right). }[/math] Then for every neighborhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ x_0 }[/math] in [math]\displaystyle{ X, }[/math] [math]\displaystyle{ y_0 }[/math] belongs to the relative interior of [math]\displaystyle{ \mathcal{R}(U) }[/math] in [math]\displaystyle{ \operatorname{aff} (\operatorname{Im} \mathcal{R}) }[/math] (i.e. [math]\displaystyle{ y_0 \in \operatorname{int}_{\operatorname{aff} (\operatorname{Im} \mathcal{R})} \mathcal{R}(U) }[/math]). In particular, if [math]\displaystyle{ {}^{ib}(\operatorname{Im} \mathcal{R}) \neq \varnothing }[/math] then [math]\displaystyle{ {}^{ib}(\operatorname{Im} \mathcal{R}) = {}^{i}(\operatorname{Im} \mathcal{R}) = \operatorname{rint} (\operatorname{Im} \mathcal{R}). }[/math]

Robinson–Ursescu theorem

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

Robinson–Ursescu theorem^[3] — Let [math]\displaystyle{ (X, \|\,\cdot\,\|) }[/math] and [math]\displaystyle{ (Y, \|\,\cdot\,\|) }[/math] be normed spaces and [math]\displaystyle{ \mathcal{R} : X \rightrightarrows Y }[/math] be a multimap with non-empty domain. Suppose that [math]\displaystyle{ Y }[/math] is a barreled space, the graph of [math]\displaystyle{ \mathcal{R} }[/math] verifies condition condition (Hwx), and that [math]\displaystyle{ (x_0, y_0) \in \operatorname{gr} \mathcal{R}. }[/math] Let [math]\displaystyle{ C_X }[/math] (resp. [math]\displaystyle{ C_Y }[/math]) denote the closed unit ball in [math]\displaystyle{ X }[/math] (resp. [math]\displaystyle{ Y }[/math]) (so [math]\displaystyle{ C_X = \{ x \in X : \| x \| \leq 1 \} }[/math]). Then the following are equivalent:

[math]\displaystyle{ y_0 }[/math] belongs to the algebraic interior of [math]\displaystyle{ \operatorname{Im} \mathcal{R}. }[/math]
[math]\displaystyle{ y_0 \in \operatorname{int} \mathcal{R}\left(x_0 + C_X\right). }[/math]
There exists [math]\displaystyle{ B \gt 0 }[/math] such that for all [math]\displaystyle{ 0 \leq r \leq 1, }[/math] [math]\displaystyle{ y_0 + B r C_Y \subseteq \mathcal{R} \left(x_0 + r C_X\right). }[/math]
There exist [math]\displaystyle{ A \gt 0 }[/math] and [math]\displaystyle{ B \gt 0 }[/math] such that for all [math]\displaystyle{ x \in x_0 + A C_X }[/math] and all [math]\displaystyle{ y \in y_0 + A C_Y, }[/math] [math]\displaystyle{ d\left(x, \mathcal{R}^{-1}(y)\right) \leq B \cdot d(y, \mathcal{R}(x)). }[/math]
There exists [math]\displaystyle{ B \gt 0 }[/math] such that for all [math]\displaystyle{ x \in X }[/math] and all [math]\displaystyle{ y \in y_0 + B C_Y, }[/math] [math]\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)). }[/math]

Notes

↑ Zălinescu 2002, p. 23.
↑ Zălinescu 2002, p. 22-23.
↑ ^3.0 ^3.1 Zălinescu 2002, p. 24.

References

Template:Zălinescu Convex Analysis in General Vector Spaces 2002
Baggs, Ivan (1974). "Functions with a closed graph" (in en). Proceedings of the American Mathematical Society 43 (2): 439–442. doi:10.1090/S0002-9939-1974-0334132-8. ISSN 0002-9939. https://www.ams.org/.

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Original source: https://en.wikipedia.org/wiki/Ursescu theorem. Read more

[FOOTNOTEZălinescu200223-1] Zălinescu 2002, p. 23.

[FOOTNOTEZălinescu200222-23-2] Zălinescu 2002, p. 22-23.

[FOOTNOTEZălinescu200224-3] 3.0 ^3.1 Zălinescu 2002, p. 24.

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Ursescu Theorem

Statement

Corollaries

Closed graph theorem

Uniform boundedness principle

Open mapping theorem

Additional corollaries

Related theorems

Simons' theorem

Robinson–Ursescu theorem

See also

Notes

References

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Ursescu theorem

Ursescu Theorem

Statement

Corollaries

Closed graph theorem

Uniform boundedness principle

Open mapping theorem

Additional corollaries

Related theorems

Simons' theorem

Robinson–Ursescu theorem

See also

Notes

References

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