Convex series

In mathematics, particularly in functional analysis and convex analysis, a {{em|{{visible anchor|convex series}}}} is a series of the form $\sum_{i=1}^{\infty} r_i x_i$ where $x_1, x_2, \ldots$ are all elements of a topological vector space $X$ , and all $r_1, r_2, \ldots$ are non-negative real numbers that sum to $1$ (that is, such that $\sum_{i=1}^{\infty} r_i = 1$ ).

Types of Convex series

Suppose that $S$ is a subset of $X$ and $\sum_{i=1}^{\infty} r_i x_i$ is a convex series in $X.$

If all $x_1, x_2, \ldots$ belong to $S$ then the convex series $\sum_{i=1}^{\infty} r_i x_i$ is called a {{visible anchor|convex series}} with elements of $S$ .
If the set $\left\{ x_1, x_2, \ldots \right\}$ is a (von Neumann) bounded set then the series called a {{visible anchor|b-convex series}}.
The convex series $\sum_{i=1}^{\infty} r_i x_i$ is said to be a {{visible anchor|convergent series}} if the sequence of partial sums $\left(\sum_{i=1}^n r_i x_i\right)_{n=1}^{\infty}$ converges in $X$ to some element of $X,$ which is called the {{visible anchor|sum of the convex series}}.
The convex series is called {{visible anchor|Cauchy}} if $\sum_{i=1}^{\infty} r_i x_i$ is a Cauchy series, which by definition means that the sequence of partial sums $\left(\sum_{i=1}^n r_i x_i\right)_{n=1}^{\infty}$ 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 $S$ is a subset of a topological vector space $X$ then $S$ is said to be a:

{{visible anchor|cs-closed set}} if any convergent convex series with elements of $S$ has its (each) sum in $S.$
In this definition, $X$ 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 $S.$
{{visible anchor|lower cs-closed set}} or a {{visible anchor|lcs-closed set}} if there exists a Fréchet space $Y$ such that $S$ is equal to the projection onto $X$ (via the canonical projection) of some cs-closed subset $B$ of $X \times Y$ 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).
{{visible anchor|ideally convex set}} if any convergent b-series with elements of $S$ has its sum in $S.$
{{visible anchor|lower ideally convex set}} or a {{visible anchor|li-convex set}} if there exists a Fréchet space $Y$ such that $S$ is equal to the projection onto $X$ (via the canonical projection) of some ideally convex subset $B$ of $X \times Y.$ Every ideally convex set is lower ideally convex. Every lower ideally convex set is convex but the converse is in general not true.
{{visible anchor|cs-complete set}} if any Cauchy convex series with elements of $S$ is convergent and its sum is in $S.$
{{visible anchor|bcs-complete set}} if any Cauchy b-convex series with elements of $S$ is convergent and its sum is in $S.$

The empty set is convex, ideally convex, bcs-complete, cs-complete, and cs-closed.

= Conditions (Hx) and (Hwx) =

If $X$ and $Y$ are topological vector spaces, $A$ is a subset of $X \times Y,$ and $x \in X$ then $A$ is said to satisfy:{{sfn|Zălinescu|2002|pp=1-23}}

{{visible anchor|Condition (Hx)}}: Whenever $\sum_{i=1}^{\infty} r_i (x_i, y_i)$ is a {{em|convex series}} with elements of $A$ such that $\sum_{i=1}^{\infty} r_i y_i$ is convergent in $Y$ with sum $y$ and $\sum_{i=1}^{\infty} r_i x_i$ is Cauchy, then $\sum_{i=1}^{\infty} r_i x_i$ is convergent in $X$ and its sum $x$ is such that $(x, y) \in A.$
{{visible anchor|Condition (Hwx)}}: Whenever $\sum_{i=1}^{\infty} r_i (x_i, y_i)$ is a {{em|b-convex series}} with elements of $A$ such that $\sum_{i=1}^{\infty} r_i y_i$ is convergent in $Y$ with sum $y$ and $\sum_{i=1}^{\infty} r_i x_i$ is Cauchy, then $\sum_{i=1}^{\infty} r_i x_i$ is convergent in $X$ and its sum $x$ is such that $(x, y) \in A.$
If X is locally convex then the statement "and $\sum_{i=1}^{\infty} r_i x_i$ is Cauchy" may be removed from the definition of condition (Hwx).

Multifunctions

The following notation and notions are used, where $\mathcal{R} : X \rightrightarrows Y$ and $\mathcal{S} : Y \rightrightarrows Z$ are multifunctions and $S \subseteq X$ is a non-empty subset of a topological vector space $X:$

The Graph of a multifunction of $\mathcal{R}$ is the set $\operatorname{gr} \mathcal{R} := \{ (x, y) \in X \times Y : y \in \mathcal{R}(x) \}.$
$\mathcal{R}$ is {{visible anchor|closed}} (respectively, {{visible anchor|cs-closed}}, {{visible anchor|lower cs-closed}}, {{visible anchor|convex}}, {{visible anchor|ideally convex}}, {{visible anchor|lower ideally convex}}, {{visible anchor|cs-complete}}, {{visible anchor|bcs-complete}}) if the same is true of the graph of $\mathcal{R}$ in $X \times Y.$
The multifunction $\mathcal{R}$ is convex if and only if for all $x_0, x_1 \in X$ and all $r \in [0, 1],$ $r \mathcal{R}\left(x_0\right) + (1 - r) \mathcal{R}\left(x_1\right) \subseteq \mathcal{R} \left(r x_0 + (1 - r) x_1\right).$
The {{visible anchor|inverse of a multifunction}} $\mathcal{R}$ is the multifunction $\mathcal{R}^{-1} : Y \rightrightarrows X$ defined by $\mathcal{R}^{-1}(y) := \left\{ x \in X : y \in \mathcal{R}(x) \right\}.$ For any subset $B \subseteq Y,$ $\mathcal{R}^{-1}(B) := \cup_{y \in B} \mathcal{R}^{-1}(y).$
The {{visible anchor|domain of a multifunction}} $\mathcal{R}$ is $\operatorname{Dom} \mathcal{R} := \left\{ x \in X : \mathcal{R}(x) \neq \emptyset \right\}.$
The {{visible anchor|image of a multifunction}} $\mathcal{R}$ is $\operatorname{Im} \mathcal{R} := \cup_{x \in X} \mathcal{R}(x).$ For any subset $A \subseteq X,$ $\mathcal{R}(A) := \cup_{x \in A} \mathcal{R}(x).$
The {{visible anchor|composition}} $\mathcal{S} \circ \mathcal{R} : X \rightrightarrows Z$ is defined by $\left(\mathcal{S} \circ \mathcal{R}\right)(x) := \cup_{y \in \mathcal{R}(x)} \mathcal{S}(y)$ for each $x \in X.$

Relationships

Let $X, Y, \text{ and } Z$ be topological vector spaces, $S \subseteq X, T \subseteq Y,$ and $A \subseteq X \times Y.$ The following implications hold:

:complete $\implies$ cs-complete $\implies$ cs-closed $\implies$ lower cs-closed (lcs-closed) {{em|and}} ideally convex.

:lower cs-closed (lcs-closed) {{em|or}} ideally convex $\implies$ lower ideally convex (li-convex) $\implies$ convex.

:(Hx) $\implies$ (Hwx) $\implies$ convex.

The converse implications do not hold in general.

If $X$ is complete then,

$S$ is cs-complete (respectively, bcs-complete) if and only if $S$ is cs-closed (respectively, ideally convex).
$A$ satisfies (Hx) if and only if $A$ is cs-closed.
$A$ satisfies (Hwx) if and only if $A$ is ideally convex.

If $Y$ is complete then,

$A$ satisfies (Hx) if and only if $A$ is cs-complete.
$A$ satisfies (Hwx) if and only if $A$ is bcs-complete.
If $B \subseteq X \times Y \times Z$ and $y \in Y$ then:
$B$ satisfies (H(x, y)) if and only if $B$ satisfies (Hx).
$B$ satisfies (Hw(x, y)) if and only if $B$ satisfies (Hwx).

If $X$ is locally convex and $\operatorname{Pr}_X (A)$ is bounded then,

If $A$ satisfies (Hx) then $\operatorname{Pr}_X (A)$ is cs-closed.
If $A$ satisfies (Hwx) then $\operatorname{Pr}_X (A)$ is ideally convex.

= Preserved properties =

Let $X_0$ be a linear subspace of $X.$ Let $\mathcal{R} : X \rightrightarrows Y$ and $\mathcal{S} : Y \rightrightarrows Z$ be multifunctions.

If $S$ is a cs-closed (resp. ideally convex) subset of $X$ then $X_0 \cap S$ is also a cs-closed (resp. ideally convex) subset of $X_0.$
If $X$ is first countable then $X_0$ is cs-closed (resp. cs-complete) if and only if $X_0$ is closed (resp. complete); moreover, if $X$ is locally convex then $X_0$ is closed if and only if $X_0$ is ideally convex.
$S \times T$ is cs-closed (resp. cs-complete, ideally convex, bcs-complete) in $X \times Y$ if and only if the same is true of both $S$ in $X$ and of $T$ in $Y.$
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 $X$ 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 $X$ 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 $X$ is a Fréchet space and the $A$ and $B$ are subsets. If $A$ and $B$ are lower ideally convex (resp. lower cs-closed) then so is $A + B.$
Suppose $X$ is a Fréchet space and $A$ is a subset of $X.$ If $A$ and $\mathcal{R} : X \rightrightarrows Y$ are lower ideally convex (resp. lower cs-closed) then so is $\mathcal{R}(A).$
Suppose $Y$ is a Fréchet space and $\mathcal{R}_2 : X \rightrightarrows Y$ is a multifunction. If $\mathcal{R}, \mathcal{R}_2, \mathcal{S}$ are all lower ideally convex (resp. lower cs-closed) then so are $\mathcal{R} + \mathcal{R}_2 : X \rightrightarrows Y$ and $\mathcal{S} \circ \mathcal{R} : X \rightrightarrows Z.$

Properties

If $S$ be a non-empty convex subset of a topological vector space $X$ then,

If $S$ is closed or open then $S$ is cs-closed.
If $X$ is Hausdorff and finite dimensional then $S$ is cs-closed.
If $X$ is first countable and $S$ is ideally convex then $\operatorname{int} S = \operatorname{int} \left(\operatorname{cl} S\right).$

Let $X$ be a Fréchet space, $Y$ be a topological vector spaces, $A \subseteq X \times Y,$ and $\operatorname{Pr}_Y : X \times Y \to Y$ be the canonical projection. If $A$ is lower ideally convex (resp. lower cs-closed) then the same is true of $\operatorname{Pr}_Y (A).$

If $X$ is a barreled first countable space and if $C \subseteq X$ then:

If $C$ is lower ideally convex then $C^i = \operatorname{int} C,$ where $C^i := \operatorname{aint}_X C$ denotes the algebraic interior of $C$ in $X.$
If $C$ is ideally convex then $C^i = \operatorname{int} C = \operatorname{int} \left(\operatorname{cl} C\right) = \left(\operatorname{cl} C\right)^i.$

Notes

References

{{Zălinescu Convex Analysis in General Vector Spaces 2002}}
{{Cite journal|last=Baggs|first=Ivan|date=1974|title=Functions with a closed graph|url=https://www.ams.org/|journal=Proceedings of the American Mathematical Society|language=en|volume=43|issue=2|pages=439–442|doi=10.1090/S0002-9939-1974-0334132-8|issn=0002-9939|doi-access=free}}

Category:Theorems in functional analysis