Convex series

In mathematics, particularly in functional analysis and convex analysis, a convex series is a series of the form $\sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i$ where $x_1,x_2,\dots$ are all elements of a topological vector space $X$, and all $r_1,r_2,\dots$ are non-negative real numbers that sum to $1$ (that is, such that $\sum _{i=1}^{\mathrm{\infty }}{r}_i=1$).

Types of Convex series

Suppose that $S$ is a subset of $X$ and $\sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i$ is a convex series in $X.$

• If all $x_1,x_2,\dots$ belong to $S$ then the convex series $\sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i$ is called a convex series with elements of $S$.
• If the set $\{x_1,x_2,\dots \}$ is a (von Neumann) bounded set then the series called a b-convex series.
• The convex series $\sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i$ is said to be a convergent series if the sequence of partial sums ${\left(\sum _{i=1}^n{r}_i{x}_i\right)}_{n=1}^{\mathrm{\infty }}$ converges in $X$ to some element of $X,$ which is called the sum of the convex series.
• The convex series is called Cauchy if $\sum _{i=1}^{\mathrm{\infty }}{r}_i{x}_i$ is a Cauchy series, which by definition means that the sequence of partial sums ${\left(\sum _{i=1}^n{r}_i{x}_i\right)}_{n=1}^{\mathrm{\infty }}$ 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:

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

Multifunctions

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

• The graph of a multifunction of ${ℛ}$ is the set $\mathrm{gr}{ℛ}:=\{(x,y)\in X\times Y:y\in {ℛ}(x)\}.$
• ${ℛ}$ 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 ${ℛ}$ in $X\times Y.$
  • The multifunction ${ℛ}$ is convex if and only if for all $x_0,x_1\in X$ and all $r\in [0,1],$ $r{ℛ}\left(x_0\right)+(1-r){ℛ}\left(x_1\right)\subseteq {ℛ}(r{x}_0+(1-r)x_1).$
• The inverse of a multifunction ${ℛ}$ is the multifunction ${{ℛ}}^{-1}:Y\rightrightarrows X$ defined by ${{ℛ}}^{-1}(y):=\{x\in X:y\in {ℛ}(x)\}.$ For any subset $B\subseteq Y,$ ${{ℛ}}^{-1}(B):={\cup }_{y\in B}{{ℛ}}^{-1}(y).$
• The domain of a multifunction ${ℛ}$ is $\mathrm{Dom}{ℛ}:=\{x\in X:{ℛ}(x)\ne \mathrm{\varnothing }\}.$
• The image of a multifunction ${ℛ}$ is $\mathrm{Im}{ℛ}:={\cup }_{x\in X}{ℛ}(x).$ For any subset $A\subseteq X,$ ${ℛ}(A):={\cup }_{x\in A}{ℛ}(x).$
• The composition ${𝒮}\circ {ℛ}:X\rightrightarrows Z$ is defined by $({𝒮}\circ {ℛ})(x):={\cup }_{y\in {ℛ}(x)}{𝒮}(y)$ 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 $⟹$ cs-complete $⟹$ cs-closed $⟹$ lower cs-closed (lcs-closed) and ideally convex.

lower cs-closed (lcs-closed) or ideally convex $⟹$ lower ideally convex (li-convex) $⟹$ convex.

(Hx) $⟹$ (Hwx) $⟹$ convex.

The converse implications do not hold in general.

If $X$ is complete then,

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

If $Y$ is complete then,

1. $A$ satisfies (Hx) if and only if $A$ is cs-complete.
2. $A$ satisfies (Hwx) if and only if $A$ is bcs-complete.
3. 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 ${\mathrm{Pr}}_X(A)$ is bounded then,

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

Preserved properties

Let $X_0$ be a linear subspace of $X.$ Let ${ℛ}:X\rightrightarrows Y$ and ${𝒮}:Y\rightrightarrows Z$ 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 ${ℛ}:X\rightrightarrows Y$ are lower ideally convex (resp. lower cs-closed) then so is ${ℛ}(A).$
• Suppose $Y$ is a Fréchet space and ${{ℛ}}_2:X\rightrightarrows Y$ is a multifunction. If ${ℛ},{{ℛ}}_2,{𝒮}$ are all lower ideally convex (resp. lower cs-closed) then so are ${ℛ}+{{ℛ}}_2:X\rightrightarrows Y$ and ${𝒮}\circ {ℛ}:X\rightrightarrows Z.$

Properties

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

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

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

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

1. If $C$ is lower ideally convex then $C^i=\mathrm{int}C,$ where $C^i:={\mathrm{aint}}_{X}C$ denotes the algebraic interior of $C$ in $X.$
2. If $C$ is ideally convex then $C^i=\mathrm{int}C=\mathrm{int}(\mathrm{cl}C)={(\mathrm{cl}C)}^i.$

See also

• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem

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/.