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:

• 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 mulifunction ${\mathcal {R}}$ 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,

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 $\operatorname {Pr} _{X}(A)$ is bounded then,

1. If $A$ satisfies (Hx) then $\operatorname {Pr} _{X}(A)$ is cs-closed.
2. 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,

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 $\operatorname {int} S=\operatorname {int} \left(\operatorname {cl} S\right).$

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:

1. 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 $X.$
2. If $C$ is ideally convex then $C^{i}=\operatorname {int} C=\operatorname {int} \left(\operatorname {cl} C\right)=\left(\operatorname {cl} C\right)^{i}.$

See also

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

References

• Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.
• Baggs, Ivan (1974). "Functions with a closed graph". Proceedings of the American Mathematical Society. 43 (2): 439–442. doi:10.1090/S0002-9939-1974-0334132-8. ISSN 0002-9939.