Convex series

In mathematics, particularly in functional analysis and convex analysis, a is a series of the form $\sum_^ r_i x_i$ 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 $\sum_^ r_i = 1$).

Types of Convex series

Suppose that $S$ is a subset of $X$ and $\sum_^ r_i x_i$ is a convex series in $X.$

* If all $x_1, x_2, \ldots$ belong to $S$ then the convex series $\sum_^ r_i x_i$ is called a with elements of $S$.
* If the set $\left\$ is a (von Neumann) bounded set then the series called a .
* The convex series $\sum_^ r_i x_i$ is said to be a if the sequence of partial sums $\left(\sum_^n r_i x_i\right)_^$ converges in $X$ to some element of $X,$ which is called the .
* The convex series is called if $\sum_^ r_i x_i$ is a Cauchy series, which by definition means that the sequence of partial sums $\left(\sum_^n r_i x_i\right)_^$ 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:

* 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 $S.$
* or a 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).
* if any convergent b-series with elements of $S$ has its sum in $S.$
* or a 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.
* if any Cauchy convex series with elements of $S$ is convergent and its sum is in $S.$
* 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:

* : Whenever $\sum_^ r_i (x_i, y_i)$ is a with elements of $A$ such that $\sum_^ r_i y_i$ is convergent in $Y$ with sum $y$ and $\sum_^ r_i x_i$ is Cauchy, then $\sum_^ r_i x_i$ is convergent in $X$ and its sum $x$ is such that $(x, y) \in A.$
* : Whenever $\sum_^ r_i (x_i, y_i)$ is a with elements of $A$ such that $\sum_^ r_i y_i$ is convergent in $Y$ with sum $y$ and $\sum_^ r_i x_i$ is Cauchy, then $\sum_^ r_i x_i$ is convergent in $X$ and its sum $x$ is such that $(x, y) \in A.$
** If X is locally convex then the statement "and $\sum_^ r_i x_i$ is Cauchy" may be removed from the definition of condition (Hw''x'').

Multifunctions

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

* The of $\mathcal$ is the set $\operatorname \mathcal := \.$
* $\mathcal$ is (respectively, , , , , , , ) if the same is true of the graph of $\mathcal$ in $X \times Y.$
** The mulifunction $\mathcal$ is convex if and only if for all $x_0, x_1 \in X$ and all r \in , 1 $r \mathcal\left(x_0\right) + (1 - r) \mathcal\left(x_1\right) \subseteq \mathcal \left(r x_0 + (1 - r) x_1\right).$
* The $\mathcal$ is the multifunction $\mathcal^ : Y \rightrightarrows X$ defined by $\mathcal^(y) := \left\.$ For any subset $B \subseteq Y,$ $\mathcal^(B) := \cup_ \mathcal^(y).$
* The $\mathcal$ is $\operatorname \mathcal := \left\.$
* The $\mathcal$ is $\operatorname \mathcal := \cup_ \mathcal(x).$ For any subset $A \subseteq X,$ $\mathcal(A) := \cup_ \mathcal(x).$
* The $\mathcal \circ \mathcal : X \rightrightarrows Z$ is defined by $\left(\mathcal \circ \mathcal\right)(x) := \cup_ \mathcal(y)$ for each $x \in X.$

Relationships

Let $X, Y, \text Z$ 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) ideally convex.
:lower cs-closed (lcs-closed) ideally convex $\implies$ lower ideally convex (li-convex) $\implies$ convex.
:(H''x'') $\implies$ (Hw''x'') $\implies$ convex.

The converse implications do not hold in general.

If $X$ is complete then,
# $S$ is cs-complete (respectively, bcs-complete) if and only if $S$ is cs-closed (respectively, ideally convex).
# $A$ satisfies (H''x'') if and only if $A$ is cs-closed.
# $A$ satisfies (Hw''x'') if and only if $A$ is ideally convex.

If $Y$ is complete then,
# $A$ satisfies (H''x'') if and only if $A$ is cs-complete.
# $A$ satisfies (Hw''x'') if and only if $A$ is bcs-complete.
# If $B \subseteq X \times Y \times Z$ and $y \in Y$ then:
## $B$ satisfies (H''(x, y)'') if and only if $B$ satisfies (H''x'').
## $B$ satisfies (Hw''(x, y)'') if and only if $B$ satisfies (Hw''x'').

If $X$ is locally convex and $\operatorname_X (A)$ is bounded then,
# If $A$ satisfies (H''x'') then $\operatorname_X (A)$ is cs-closed.
# If $A$ satisfies (Hw''x'') then $\operatorname_X (A)$ is ideally convex.

Preserved properties

Let $X_0$ be a linear subspace of $X.$ Let $\mathcal : X \rightrightarrows Y$ and $\mathcal : Y \rightrightarrows Z$ 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 $\mathcal : X \rightrightarrows Y$ are lower ideally convex (resp. lower cs-closed) then so is $\mathcal(A).$
* Suppose $Y$ is a Fréchet space and $\mathcal_2 : X \rightrightarrows Y$ is a multifunction. If $\mathcal, \mathcal_2, \mathcal$ are all lower ideally convex (resp. lower cs-closed) then so are $\mathcal + \mathcal_2 : X \rightrightarrows Y$ and $\mathcal \circ \mathcal : X \rightrightarrows Z.$

Properties

If $S$ 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 $\operatorname S = \operatorname \left(\operatorname S\right).$

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

If $X$ is a barreled first countable space and if $C \subseteq X$ then:
# If $C$ is lower ideally convex then $C^i = \operatorname C,$ where $C^i := \operatorname_X C$ denotes the algebraic interior of $C$ in $X.$
# If $C$ is ideally convex then $C^i = \operatorname C = \operatorname \left(\operatorname C\right) = \left(\operatorname C\right)^i.$

See also

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Notes

References

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{{Analysis in topological vector spaces

Theorems in functional analysis