Convex series

In mathematics, particularly in functional analysis and convex analysis, a convex series is a series of the form ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$ where ${\displaystyle x_{1},x_{2},\ldots }$ are all elements of a topological vector space X, all ${\displaystyle r_{1},r_{2},\ldots }$ are non-negative real numbers that sum to 1 (i.e. ${\displaystyle \sum _{i=1}^{\infty }r_{i}=1}$).

Types of Convex series

Suppose that S is a subset of X and ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$ is a convex series in X.

• If all ${\displaystyle x_{1},x_{2},\ldots }$ belong to S then we say that the convex series ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$ is a convex series with elements of S.
• If the set ${\displaystyle \{x_{1},x_{2},\ldots \}}$ is von Neumann bounded then we call the series a b-convex series.
• We say that the convex series is convergent if ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$ converges in X (i.e. if the sequence of partial sums ${\displaystyle \left(\sum _{i=1}^{n}r_{i}x_{i}\right)_{n=1}^{\infty }}$ converges in X to some element of X, which is called the convex series' sum).
• We call the convex series Cauchy if ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$ is a Cauchy series (i.e. if the sequence of partial sums ${\displaystyle \left(\sum _{i=1}^{n}r_{i}x_{i}\right)_{n=1}^{\infty }}$ 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.

Let S be a subset of a topological vector space X. Then say that S is:

• cs-closed if any convergent convex series with elements of S has its (each) sum in S.
  • Note that 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 or lcs-closed 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 ${\displaystyle X\times Y}$ 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 if any convergent b-series with elements of S has its sum in S.
• lower ideally convex or li-convex 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 ${\displaystyle X\times Y}$. 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 if any Cauchy convex series with elements of S is convergent and its sum is in S.
• bcs-complete if any Cauchy b-convex series with elements of S is convergent and its sum is in S.

Note that 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 ${\displaystyle X\times Y}$, and x is an element of X then we say that A satisfies:

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

Multifunctions

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

• The graph of ${\displaystyle {\mathcal {R}}}$ is ${\displaystyle \operatorname {gr} {\mathcal {R}}:=\left\{(x,y)\in X\times Y:y\in {\mathcal {R}}(x)\right\}}$.
• ${\displaystyle {\mathcal {R}}}$ 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 ${\displaystyle {\mathcal {R}}}$ in ${\displaystyle X\times Y}$.
  • Note that ${\displaystyle {\mathcal {R}}}$ is convex if and only if for all ${\displaystyle x_{0},x_{1}\in X}$ and all ${\displaystyle r\in [0,1]}$, ${\displaystyle r{\mathcal {R}}\left(x_{0}\right)+(1-r){\mathcal {R}}\left(x_{1}\right)\subseteq {\mathcal {R}}\left(rx_{0}+(1-r)x_{1}\right)}$.
• The inverse of ${\displaystyle {\mathcal {R}}}$ is the multifunction ${\displaystyle {\mathcal {R}}^{-1}:Y\rightrightarrows X}$ defined by ${\displaystyle {\mathcal {R}}^{-1}(y):=\left\{x\in X:y\in {\mathcal {R}}(x)\right\}}$. For any subset ${\displaystyle B\subseteq Y}$, ${\displaystyle {\mathcal {R}}^{-1}(B):=\cup _{y\in B}{\mathcal {R}}^{-1}(y)}$.
• The domain of ${\displaystyle {\mathcal {R}}}$ is ${\displaystyle \operatorname {Dom} {\mathcal {R}}:=\left\{x\in X:{\mathcal {R}}(x)\neq \emptyset \right\}}$.
• The image of ${\displaystyle {\mathcal {R}}}$ is ${\displaystyle \operatorname {Im} {\mathcal {R}}:=\cup _{x\in X}{\mathcal {R}}(x)}$. For any subset ${\displaystyle A\subseteq X}$, ${\displaystyle {\mathcal {R}}(A):=\cup _{x\in A}{\mathcal {R}}(x)}$.
• The composition ${\displaystyle {\mathcal {S}}\circ {\mathcal {R}}:X\rightrightarrows Z}$ is defined by ${\displaystyle \left({\mathcal {S}}\circ {\mathcal {R}}\right)(x):=\cup _{y\in {\mathcal {R}}(x)}{\mathcal {S}}(y)}$ for each ${\displaystyle x\in X}$.

Relationships

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

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

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

(Hx) ${\displaystyle \implies }$ (Hwx) ${\displaystyle \implies }$ convex.

The converse implication do not hold in general.

If X is complete then,

1. S is cs-complete (resp. bcs-complete) if and only if S is cs-closed (resp. 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 ${\displaystyle B\subseteq X\times Y\times Z}$ and ${\displaystyle y\in Y}$ 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 ${\displaystyle \operatorname {Pr} _{X}(A)}$ is bounded then,

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

Preserved properties

Let ${\displaystyle X_{0}}$ be a linear subspace of X. Let ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}$ and ${\displaystyle {\mathcal {S}}:Y\rightrightarrows Z}$ be multifunctions.

• If S is a cs-closed (resp. ideally convex) subset of X then ${\displaystyle X_{0}\cap S}$ is also a cs-closed (resp. ideally convex) subset of ${\displaystyle X_{0}}$.
• If X is first countable then ${\displaystyle X_{0}}$ is cs-closed (resp. cs-complete) if and only if ${\displaystyle X_{0}}$ is closed (resp. complete); moreover, if X is locally convex then ${\displaystyle X_{0}}$ is closed if and only if ${\displaystyle X_{0}}$ is ideally convex.
• ${\displaystyle S\times T}$ is cs-closed (resp. cs-complete, ideally convex, bcs-complete) in ${\displaystyle X\times Y}$ 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 ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}$ are lower ideally convex (resp. lower cs-closed) then so is ${\displaystyle {\mathcal {R}}(A)}$.
• Suppose Y is a Fréchet space and ${\displaystyle {\mathcal {R}}_{2}:X\rightrightarrows Y}$ is a multifunction. If ${\displaystyle {\mathcal {R}},{\mathcal {R}}_{2},{\mathcal {S}}}$ are all lower ideally convex (resp. lower cs-closed) then so are ${\displaystyle {\mathcal {R}}+{\mathcal {R}}_{2}:X\rightrightarrows Y}$ and ${\displaystyle {\mathcal {S}}\circ {\mathcal {R}}:X\rightrightarrows Z}$.

Properties

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

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

Let X be a barreled first countable space and let C be a subset of X. Then:

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

See also

• Ursescu theorem

Notes

References

• Zalinescu, C (2002). Convex analysis in general vector spaces. River Edge, N.J. London: World Scientific. ISBN 981-238-067-1. OCLC 285163112.CS1 maint: ref=harv (link)
• 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.