Convex series
In mathematics, particularly in functional analysis and convex analysis, a convex series is a series of the form [math]\displaystyle{ \sum_{i=1}^{\infty} r_i x_i }[/math] where [math]\displaystyle{ x_1, x_2, \ldots }[/math] are all elements of a topological vector space [math]\displaystyle{ X }[/math], and all [math]\displaystyle{ r_1, r_2, \ldots }[/math] are non-negative real numbers that sum to [math]\displaystyle{ 1 }[/math] (that is, such that [math]\displaystyle{ \sum_{i=1}^{\infty} r_i = 1 }[/math]).
Types of Convex series
Suppose that [math]\displaystyle{ S }[/math] is a subset of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \sum_{i=1}^{\infty} r_i x_i }[/math] is a convex series in [math]\displaystyle{ X. }[/math]
- If all [math]\displaystyle{ x_1, x_2, \ldots }[/math] belong to [math]\displaystyle{ S }[/math] then the convex series [math]\displaystyle{ \sum_{i=1}^{\infty} r_i x_i }[/math] is called a convex series with elements of [math]\displaystyle{ S }[/math].
- If the set [math]\displaystyle{ \left\{ x_1, x_2, \ldots \right\} }[/math] is a (von Neumann) bounded set then the series called a b-convex series.
- The convex series [math]\displaystyle{ \sum_{i=1}^{\infty} r_i x_i }[/math] is said to be a convergent series if the sequence of partial sums [math]\displaystyle{ \left(\sum_{i=1}^n r_i x_i\right)_{n=1}^{\infty} }[/math] converges in [math]\displaystyle{ X }[/math] to some element of [math]\displaystyle{ X, }[/math] which is called the sum of the convex series.
- The convex series is called Cauchy if [math]\displaystyle{ \sum_{i=1}^{\infty} r_i x_i }[/math] is a Cauchy series, which by definition means that the sequence of partial sums [math]\displaystyle{ \left(\sum_{i=1}^n r_i x_i\right)_{n=1}^{\infty} }[/math] is a Cauchy sequence.
Types of subsets
Convex series allow for the definition of special types of subsets that are well-behaved and useful with very good stability properties.
If [math]\displaystyle{ S }[/math] is a subset of a topological vector space [math]\displaystyle{ X }[/math] then [math]\displaystyle{ S }[/math] is said to be a:
- cs-closed set if any convergent convex series with elements of [math]\displaystyle{ S }[/math] has its (each) sum in [math]\displaystyle{ S. }[/math]
- In this definition, [math]\displaystyle{ X }[/math] is not required to be Hausdorff, in which case the sum may not be unique. In any such case we require that every sum belong to [math]\displaystyle{ S. }[/math]
- lower cs-closed set or a lcs-closed set 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] (via the canonical projection) of some cs-closed subset [math]\displaystyle{ B }[/math] of [math]\displaystyle{ X \times Y }[/math] Every cs-closed set is lower cs-closed and every lower cs-closed set is lower ideally convex and convex (the converses are not true in general).
- ideally convex set if any convergent b-series with elements of [math]\displaystyle{ S }[/math] has its sum in [math]\displaystyle{ S. }[/math]
- lower ideally convex set or a li-convex set 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] (via the canonical projection) of some ideally convex subset [math]\displaystyle{ B }[/math] of [math]\displaystyle{ X \times Y. }[/math] Every ideally convex set is lower ideally convex. Every lower ideally convex set is convex but the converse is in general not true.
- cs-complete set if any Cauchy convex series with elements of [math]\displaystyle{ S }[/math] is convergent and its sum is in [math]\displaystyle{ S. }[/math]
- bcs-complete set if any Cauchy b-convex series with elements of [math]\displaystyle{ S }[/math] is convergent and its sum is in [math]\displaystyle{ S. }[/math]
The empty set is convex, ideally convex, bcs-complete, cs-complete, and cs-closed.
Conditions (Hx) and (Hwx)
If [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are topological vector spaces, [math]\displaystyle{ A }[/math] is a subset of [math]\displaystyle{ X \times Y, }[/math] and [math]\displaystyle{ x \in X }[/math] then [math]\displaystyle{ A }[/math] is said to satisfy:[1]
- Condition (Hx): Whenever [math]\displaystyle{ \sum_{i=1}^{\infty} r_i (x_i, y_i) }[/math] is a convex series with elements of [math]\displaystyle{ A }[/math] such that [math]\displaystyle{ \sum_{i=1}^{\infty} r_i y_i }[/math] is convergent in [math]\displaystyle{ Y }[/math] with sum [math]\displaystyle{ y }[/math] and [math]\displaystyle{ \sum_{i=1}^{\infty} r_i x_i }[/math] is Cauchy, then [math]\displaystyle{ \sum_{i=1}^{\infty} r_i x_i }[/math] is convergent in [math]\displaystyle{ X }[/math] and its sum [math]\displaystyle{ x }[/math] is such that [math]\displaystyle{ (x, y) \in A. }[/math]
- Condition (Hwx): Whenever [math]\displaystyle{ \sum_{i=1}^{\infty} r_i (x_i, y_i) }[/math] is a b-convex series with elements of [math]\displaystyle{ A }[/math] such that [math]\displaystyle{ \sum_{i=1}^{\infty} r_i y_i }[/math] is convergent in [math]\displaystyle{ Y }[/math] with sum [math]\displaystyle{ y }[/math] and [math]\displaystyle{ \sum_{i=1}^{\infty} r_i x_i }[/math] is Cauchy, then [math]\displaystyle{ \sum_{i=1}^{\infty} r_i x_i }[/math] is convergent in [math]\displaystyle{ X }[/math] and its sum [math]\displaystyle{ x }[/math] is such that [math]\displaystyle{ (x, y) \in A. }[/math]
- If X is locally convex then the statement "and [math]\displaystyle{ \sum_{i=1}^{\infty} r_i x_i }[/math] is Cauchy" may be removed from the definition of condition (Hwx).
Multifunctions
The following notation and notions are used, where [math]\displaystyle{ \mathcal{R} : X \rightrightarrows Y }[/math] and [math]\displaystyle{ \mathcal{S} : Y \rightrightarrows Z }[/math] are multifunctions and [math]\displaystyle{ S \subseteq X }[/math] is a non-empty subset of a topological vector space [math]\displaystyle{ X: }[/math]
- The graph of a multifunction of [math]\displaystyle{ \mathcal{R} }[/math] is the set [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, cs-closed, lower cs-closed, convex, ideally convex, lower ideally convex, cs-complete, bcs-complete) if the same is true of the graph of [math]\displaystyle{ \mathcal{R} }[/math] in [math]\displaystyle{ X \times Y. }[/math]
- The mulifunction [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 a multifunction [math]\displaystyle{ \mathcal{R} }[/math] is the multifunction [math]\displaystyle{ \mathcal{R}^{-1} : Y \rightrightarrows X }[/math] defined by [math]\displaystyle{ \mathcal{R}^{-1}(y) := \left\{ x \in X : y \in \mathcal{R}(x) \right\}. }[/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]
- The domain of a multifunction [math]\displaystyle{ \mathcal{R} }[/math] is [math]\displaystyle{ \operatorname{Dom} \mathcal{R} := \left\{ x \in X : \mathcal{R}(x) \neq \emptyset \right\}. }[/math]
- The image of a multifunction [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 composition [math]\displaystyle{ \mathcal{S} \circ \mathcal{R} : X \rightrightarrows Z }[/math] is defined by [math]\displaystyle{ \left(\mathcal{S} \circ \mathcal{R}\right)(x) := \cup_{y \in \mathcal{R}(x)} \mathcal{S}(y) }[/math] for each [math]\displaystyle{ x \in X. }[/math]
Relationships
Let [math]\displaystyle{ X, Y, \text{ and } Z }[/math] be topological vector spaces, [math]\displaystyle{ S \subseteq X, T \subseteq Y, }[/math] and [math]\displaystyle{ A \subseteq X \times Y. }[/math] The following implications hold:
- complete [math]\displaystyle{ \implies }[/math] cs-complete [math]\displaystyle{ \implies }[/math] cs-closed [math]\displaystyle{ \implies }[/math] lower cs-closed (lcs-closed) and ideally convex.
- lower cs-closed (lcs-closed) or ideally convex [math]\displaystyle{ \implies }[/math] lower ideally convex (li-convex) [math]\displaystyle{ \implies }[/math] convex.
- (Hx) [math]\displaystyle{ \implies }[/math] (Hwx) [math]\displaystyle{ \implies }[/math] convex.
The converse implications do not hold in general.
If [math]\displaystyle{ X }[/math] is complete then,
- [math]\displaystyle{ S }[/math] is cs-complete (respectively, bcs-complete) if and only if [math]\displaystyle{ S }[/math] is cs-closed (respectively, ideally convex).
- [math]\displaystyle{ A }[/math] satisfies (Hx) if and only if [math]\displaystyle{ A }[/math] is cs-closed.
- [math]\displaystyle{ A }[/math] satisfies (Hwx) if and only if [math]\displaystyle{ A }[/math] is ideally convex.
If [math]\displaystyle{ Y }[/math] is complete then,
- [math]\displaystyle{ A }[/math] satisfies (Hx) if and only if [math]\displaystyle{ A }[/math] is cs-complete.
- [math]\displaystyle{ A }[/math] satisfies (Hwx) if and only if [math]\displaystyle{ A }[/math] is bcs-complete.
- If [math]\displaystyle{ B \subseteq X \times Y \times Z }[/math] and [math]\displaystyle{ y \in Y }[/math] then:
- [math]\displaystyle{ B }[/math] satisfies (H(x, y)) if and only if [math]\displaystyle{ B }[/math] satisfies (Hx).
- [math]\displaystyle{ B }[/math] satisfies (Hw(x, y)) if and only if [math]\displaystyle{ B }[/math] satisfies (Hwx).
If [math]\displaystyle{ X }[/math] is locally convex and [math]\displaystyle{ \operatorname{Pr}_X (A) }[/math] is bounded then,
- If [math]\displaystyle{ A }[/math] satisfies (Hx) then [math]\displaystyle{ \operatorname{Pr}_X (A) }[/math] is cs-closed.
- If [math]\displaystyle{ A }[/math] satisfies (Hwx) then [math]\displaystyle{ \operatorname{Pr}_X (A) }[/math] is ideally convex.
Preserved properties
Let [math]\displaystyle{ X_0 }[/math] be a linear subspace of [math]\displaystyle{ X. }[/math] Let [math]\displaystyle{ \mathcal{R} : X \rightrightarrows Y }[/math] and [math]\displaystyle{ \mathcal{S} : Y \rightrightarrows Z }[/math] be multifunctions.
- If [math]\displaystyle{ S }[/math] is a cs-closed (resp. ideally convex) subset of [math]\displaystyle{ X }[/math] then [math]\displaystyle{ X_0 \cap S }[/math] is also a cs-closed (resp. ideally convex) subset of [math]\displaystyle{ X_0. }[/math]
- If [math]\displaystyle{ X }[/math] is first countable then [math]\displaystyle{ X_0 }[/math] is cs-closed (resp. cs-complete) if and only if [math]\displaystyle{ X_0 }[/math] is closed (resp. complete); moreover, if [math]\displaystyle{ X }[/math] is locally convex then [math]\displaystyle{ X_0 }[/math] is closed if and only if [math]\displaystyle{ X_0 }[/math] is ideally convex.
- [math]\displaystyle{ S \times T }[/math] is cs-closed (resp. cs-complete, ideally convex, bcs-complete) in [math]\displaystyle{ X \times Y }[/math] if and only if the same is true of both [math]\displaystyle{ S }[/math] in [math]\displaystyle{ X }[/math] and of [math]\displaystyle{ T }[/math] in [math]\displaystyle{ Y. }[/math]
- The properties of being cs-closed, lower cs-closed, ideally convex, lower ideally convex, cs-complete, and bcs-complete are all preserved under isomorphisms of topological vector spaces.
- The intersection of arbitrarily many cs-closed (resp. ideally convex) subsets of [math]\displaystyle{ X }[/math] has the same property.
- The Cartesian product of cs-closed (resp. ideally convex) subsets of arbitrarily many topological vector spaces has that same property (in the product space endowed with the product topology).
- The intersection of countably many lower ideally convex (resp. lower cs-closed) subsets of [math]\displaystyle{ X }[/math] has the same property.
- The Cartesian product of lower ideally convex (resp. lower cs-closed) subsets of countably many topological vector spaces has that same property (in the product space endowed with the product topology).
- Suppose [math]\displaystyle{ X }[/math] is a Fréchet space and the [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] are subsets. If [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] are lower ideally convex (resp. lower cs-closed) then so is [math]\displaystyle{ A + B. }[/math]
- Suppose [math]\displaystyle{ X }[/math] is a Fréchet space and [math]\displaystyle{ A }[/math] is a subset of [math]\displaystyle{ X. }[/math] If [math]\displaystyle{ A }[/math] and [math]\displaystyle{ \mathcal{R} : X \rightrightarrows Y }[/math] are lower ideally convex (resp. lower cs-closed) then so is [math]\displaystyle{ \mathcal{R}(A). }[/math]
- Suppose [math]\displaystyle{ Y }[/math] is a Fréchet space and [math]\displaystyle{ \mathcal{R}_2 : X \rightrightarrows Y }[/math] is a multifunction. If [math]\displaystyle{ \mathcal{R}, \mathcal{R}_2, \mathcal{S} }[/math] are all lower ideally convex (resp. lower cs-closed) then so are [math]\displaystyle{ \mathcal{R} + \mathcal{R}_2 : X \rightrightarrows Y }[/math] and [math]\displaystyle{ \mathcal{S} \circ \mathcal{R} : X \rightrightarrows Z. }[/math]
Properties
If [math]\displaystyle{ S }[/math] be a non-empty convex subset of a topological vector space [math]\displaystyle{ X }[/math] then,
- If [math]\displaystyle{ S }[/math] is closed or open then [math]\displaystyle{ S }[/math] is cs-closed.
- If [math]\displaystyle{ X }[/math] is Hausdorff and finite dimensional then [math]\displaystyle{ S }[/math] is cs-closed.
- If [math]\displaystyle{ X }[/math] is first countable and [math]\displaystyle{ S }[/math] is ideally convex then [math]\displaystyle{ \operatorname{int} S = \operatorname{int} \left(\operatorname{cl} S\right). }[/math]
Let [math]\displaystyle{ X }[/math] be a Fréchet space, [math]\displaystyle{ Y }[/math] be a topological vector spaces, [math]\displaystyle{ A \subseteq X \times Y, }[/math] and [math]\displaystyle{ \operatorname{Pr}_Y : X \times Y \to Y }[/math] be the canonical projection. If [math]\displaystyle{ A }[/math] is lower ideally convex (resp. lower cs-closed) then the same is true of [math]\displaystyle{ \operatorname{Pr}_Y (A). }[/math]
If [math]\displaystyle{ X }[/math] is a barreled first countable space and if [math]\displaystyle{ C \subseteq X }[/math] then:
- If [math]\displaystyle{ C }[/math] is lower ideally convex then [math]\displaystyle{ C^i = \operatorname{int} C, }[/math] where [math]\displaystyle{ C^i := \operatorname{aint}_X C }[/math] denotes the algebraic interior of [math]\displaystyle{ C }[/math] in [math]\displaystyle{ X. }[/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]
See also
- Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
Notes
- ↑ Zălinescu 2002, pp. 1-23.
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/.
Original source: https://en.wikipedia.org/wiki/Convex series.
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