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

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 we say that the convex series $\sum _{i=1}^{\infty }r_{i}x_{i}$ is a convex series with elements of S.
If the set $\{x_{1},x_{2},\ldots \}$ is von Neumann bounded then we call the series a b-convex series.
We say that the convex series is convergent if $\sum _{i=1}^{\infty }r_{i}x_{i}$ converges in X (i.e. 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 convex series' sum).
We call the convex series Cauchy if $\sum _{i=1}^{\infty }r_{i}x_{i}$ is a Cauchy series (i.e. if 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.

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 $X\times Y$ 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 $X\times Y$ . 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 $X\times Y$ , and x is an element of X then we say that A satisfies:

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 $\sum _{i=1}^{\infty }r_{i}(x_{i},y_{i})$ $\text{[math]}$ is a b-convex series with elements of A such that $\sum _{i=1}^{\infty }r_{i}y_{i}$ $\text{[math]}$ is convergent in Y with sum y and $\sum _{i=1}^{\infty }r_{i}x_{i}$ $\text{[math]}$ is Cauchy, then $\sum _{i=1}^{\infty }r_{i}x_{i}$ $\text{[math]}$ is convergent in X and its sum x is such that $(x,y)\in A$ $\text{[math]}$ .
- Note that 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 ${\mathcal {S}}:Y\rightrightarrows Z$ be multifunctions and S is a non-empty subset of a topological vector space X:

The graph of ${\mathcal {R}}$ is $\operatorname {gr} {\mathcal {R}}:=\left\{(x,y)\in X\times Y:y\in {\mathcal {R}}(x)\right\}$ .
${\mathcal {R}}$ $\text{[math]}$ 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 ${\mathcal {R}}$ $\text{[math]}$ in $X\times Y$ $\text{[math]}$ .
- Note that ${\mathcal {R}}$ 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 ${\mathcal {R}}$ 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 ${\mathcal {R}}$ is $\operatorname {Dom} {\mathcal {R}}:=\left\{x\in X:{\mathcal {R}}(x)\neq \emptyset \right\}$ .
The image of ${\mathcal {R}}$ 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, 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 implication do not hold in general.

If X is complete then,

S is cs-complete (resp. bcs-complete) if and only if S is cs-closed (resp. 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 $B\subseteq X\times Y\times Z$ $\text{[math]}$ and $y\in Y$ $\text{[math]}$ 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

Let 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 $\operatorname {Pr} _{Y}(A)$ .

Let X be a barreled first countable space and let C be a subset of 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 $C^{i}=\operatorname {int} C=\operatorname {int} \left(\operatorname {cl} C\right)=\left(\operatorname {cl} C\right)^{i}$ .

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.

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Functional analysis (topics, glossary)
Spaces	Hilbert space, Banach space, Fréchet space, topological vector space
Theorems	Hahn–Banach theorem, closed graph theorem, uniform boundedness principle, Kakutani fixed-point theorem, min-max theorem, Gelfand–Naimark theorem, Banach–Alaoglu theorem
Operators	bounded operator, compact operator, adjoint operator, unitary operator, Hilbert–Schmidt operator, trace class, unbounded operator
Algebras	Banach algebra, C-algebra, spectrum of a C-algebra, operator algebra, group algebra of a locally compact group, von Neumann algebra
Open problems	invariant subspace problem, Mahler's conjecture
Applications	Besov space, Hardy space, spectral theory of ordinary differential equations, heat kernel, index theorem, calculus of variation, functional calculus, integral operator, Jones polynomial, topological quantum field theory, noncommutative geometry
Advanced topics	locally convex space, approximation property, balanced set, Schwartz space, weak topology, barrelled space, Banach–Mazur distance, Tomita–Takesaki theory

Convex analysis
Main results	Mazur's lemma Robinson-Ursescu Simons Ursescu
Types of sets	Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Zonotope

Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Analysis of vector-valued curves Bochner space Convex series
Derivatives	Differentiable vector-valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet Gateaux functional holomorphic quasi
Measurability	Measures (Lebesgue Projection-valued Vector) Weakly / Strongly measurable function
Integrals	Bochner Dunford Pettis/Gelfand–Pettis/Weak regulated Paley–Wiener
Main results	Inverse function theorem (Nash–Moser theorem)
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus