Convex series

In mathematics, particularly in functional analysis and convex analysis, a 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 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 b-convex series.
The convex series $\sum _{i=1}^{\infty }r_{i}x_{i}$ is said to be a 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 sum of the convex series.
The convex series is called 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:

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 $S.$
lower cs-closed set or a 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).
ideally convex set if any convergent b-series with elements of $S$ has its sum in $S.$
lower ideally convex set or a 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.
cs-complete set if any Cauchy convex series with elements of $S$ is convergent and its sum is in $S.$
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:^[1]

Condition (Hx): Whenever $\sum _{i=1}^{\infty }r_{i}(x_{i},y_{i})$ is a 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 $(x,y)\in A.$
Condition (Hwx): Whenever $\sum _{i=1}^{\infty }r_{i}(x_{i},y_{i})$ is a 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 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 ${\mathcal {R}}$ in $X\times Y.$
- The multifunction ${\mathcal {R}}$ is convex if and only if for all $x_{0},x_{1}\in X$ and all $r\in [0,1],$ $r{\mathcal {R}}\left(x_{0}\right)+(1-r){\mathcal {R}}\left(x_{1}\right)\subseteq {\mathcal {R}}\left(rx_{0}+(1-r)x_{1}\right).$
The 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 domain of a multifunction ${\mathcal {R}}$ is $\operatorname {Dom} {\mathcal {R}}:=\left\{x\in X:{\mathcal {R}}(x)\neq \emptyset \right\}.$
The 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 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) and ideally convex.

lower cs-closed (lcs-closed) 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:
1. $B$ satisfies (H(x, y)) if and only if $B$ satisfies (Hx).
2. $B$ 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 $C^{i}=\operatorname {int} C=\operatorname {int} \left(\operatorname {cl} C\right)=\left(\operatorname {cl} C\right)^{i}.$