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 ${\displaystyle X}$, and all ${\displaystyle r_{1},r_{2},\ldots }$ are non-negative real numbers that sum to ${\displaystyle 1}$ (that is, such that ${\displaystyle \sum _{i=1}^{\infty }r_{i}=1}$).

Types of Convex series

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

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

If ${\displaystyle S}$ is a subset of a topological vector space ${\displaystyle X}$ then ${\displaystyle S}$ is said to be a:

• cs-closed set if any convergent convex series with elements of ${\displaystyle S}$ has its (each) sum in ${\displaystyle S.}$
  • In this definition, ${\displaystyle X}$ 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 ${\displaystyle S.}$
• lower cs-closed set or a lcs-closed set if there exists a Fréchet space ${\displaystyle Y}$ such that ${\displaystyle S}$ is equal to the projection onto ${\displaystyle X}$ (via the canonical projection) of some cs-closed subset ${\displaystyle B}$ 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 set if any convergent b-series with elements of ${\displaystyle S}$ has its sum in ${\displaystyle S.}$
• lower ideally convex set or a li-convex set if there exists a Fréchet space ${\displaystyle Y}$ such that ${\displaystyle S}$ is equal to the projection onto ${\displaystyle X}$ (via the canonical projection) of some ideally convex subset ${\displaystyle B}$ 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 set if any Cauchy convex series with elements of ${\displaystyle S}$ is convergent and its sum is in ${\displaystyle S.}$
• bcs-complete set if any Cauchy b-convex series with elements of ${\displaystyle S}$ is convergent and its sum is in ${\displaystyle S.}$

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

Conditions (Hx) and (Hwx)

If ${\displaystyle X}$ and ${\displaystyle Y}$ are topological vector spaces, ${\displaystyle A}$ is a subset of ${\displaystyle X\times Y,}$ and ${\displaystyle x\in X}$ then ${\displaystyle A}$ is said to satisfy: ^[1]

• Condition (Hx): Whenever ${\displaystyle \sum _{i=1}^{\infty }r_{i}(x_{i},y_{i})}$ is a convex series with elements of ${\displaystyle A}$ such that ${\displaystyle \sum _{i=1}^{\infty }r_{i}y_{i}}$ is convergent in ${\displaystyle Y}$ with sum ${\displaystyle y}$ 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 ${\displaystyle X}$ and its sum ${\displaystyle x}$ 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 ${\displaystyle A}$ such that ${\displaystyle \sum _{i=1}^{\infty }r_{i}y_{i}}$ is convergent in ${\displaystyle Y}$ with sum ${\displaystyle y}$ 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 ${\displaystyle X}$ and its sum ${\displaystyle x}$ is such that ${\displaystyle (x,y)\in A.}$
  • 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 ${\displaystyle {\mathcal {S}}:Y\rightrightarrows Z}$ are multifunctions and ${\displaystyle S\subseteq X}$ is a non-empty subset of a topological vector space ${\displaystyle X:}$

• The graph of a multifunction of ${\displaystyle {\mathcal {R}}}$ is the set ${\displaystyle \operatorname {gr} {\mathcal {R}}:=\{(x,y)\in X\times Y:y\in {\mathcal {R}}(x)\}.}$
• ${\displaystyle {\mathcal {R}}}$ 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 ${\displaystyle X\times Y.}$
  • The multifunction ${\displaystyle {\mathcal {R}}}$ is convex if and only if for all ${\displaystyle x_{0},x_{1}\in X}$ and all ${\displaystyle r\in [0,1],}$${\displaystyle r{\mathcal {R}}\left(x_{0}\right)+(1-r){\mathcal {R}}\left(x_{1}\right)\subseteq {\mathcal {R}}\left(rx_{0}+(1-r)x_{1}\right).}$
• The inverse of a multifunction${\displaystyle {\mathcal {R}}}$ is the multifunction ${\displaystyle {\mathcal {R}}^{-1}:Y\rightrightarrows X}$ defined by ${\displaystyle {\mathcal {R}}^{-1}(y):=\left\{x\in X:y\in {\mathcal {R}}(x)\right\}.}$ For any subset ${\displaystyle B\subseteq Y,}$${\displaystyle {\mathcal {R}}^{-1}(B):=\cup _{y\in B}{\mathcal {R}}^{-1}(y).}$
• The domain of a multifunction${\displaystyle {\mathcal {R}}}$ is ${\displaystyle \operatorname {Dom} {\mathcal {R}}:=\left\{x\in X:{\mathcal {R}}(x)\neq \emptyset \right\}.}$
• The image of a multifunction${\displaystyle {\mathcal {R}}}$ 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 ${\displaystyle X,Y,{\text{ and }}Z}$ be topological vector spaces, ${\displaystyle S\subseteq X,T\subseteq Y,}$ 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 implications do not hold in general.

If ${\displaystyle X}$ is complete then,

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

If ${\displaystyle Y}$ is complete then,

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

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

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

Preserved properties

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

• If ${\displaystyle S}$ is a cs-closed (resp. ideally convex) subset of ${\displaystyle X}$ then ${\displaystyle X_{0}\cap S}$ is also a cs-closed (resp. ideally convex) subset of ${\displaystyle X_{0}.}$
• If ${\displaystyle X}$ 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 ${\displaystyle X}$ 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 ${\displaystyle S}$ in ${\displaystyle X}$ and of ${\displaystyle T}$ in ${\displaystyle Y.}$
• 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 ${\displaystyle X}$ 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 ${\displaystyle X}$ 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 ${\displaystyle X}$ is a Fréchet space and the ${\displaystyle A}$ and ${\displaystyle B}$ are subsets. If ${\displaystyle A}$ and ${\displaystyle B}$ are lower ideally convex (resp. lower cs-closed) then so is ${\displaystyle A+B.}$
• Suppose ${\displaystyle X}$ is a Fréchet space and ${\displaystyle A}$ is a subset of ${\displaystyle X.}$ If ${\displaystyle A}$ and ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}$ are lower ideally convex (resp. lower cs-closed) then so is ${\displaystyle {\mathcal {R}}(A).}$
• Suppose ${\displaystyle Y}$ 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

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

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

Let ${\displaystyle X}$ be a Fréchet space, ${\displaystyle Y}$ 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 ${\displaystyle A}$ is lower ideally convex (resp. lower cs-closed) then the same is true of ${\displaystyle \operatorname {Pr} _{Y}(A).}$

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

1. If ${\displaystyle C}$ is lower ideally convex then ${\displaystyle C^{i}=\operatorname {int} C,}$ where ${\displaystyle C^{i}:=\operatorname {aint} _{X}C}$ denotes the algebraic interior of ${\displaystyle C}$ in ${\displaystyle X.}$
2. If ${\displaystyle C}$ 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  – Generalization of closed graph, open mapping, and uniform boundedness theorem

Notes

1. Zălinescu 2002, pp. 1–23.

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.