Convex series

In mathematics, particularly in functional analysis and convex analysis, a 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  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 .
 * The convex series $$\sum_{i=1}^{\infty} r_i x_i$$ is said to be a ' 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 '.
 * The convex series is called  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:


 *  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 $$\sum_{i=1}^{\infty} r_i (x_i, y_i)$$ is a 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 $$(x, y) \in A.$$
 * : Whenever $$\sum_{i=1}^{\infty} r_i (x_i, y_i)$$ is a 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 ' (respectively, ', ', ', ', ', ', ') 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 $$r \in [0, 1],$$ $$r \mathcal{R}\left(x_0\right) + (1 - r) \mathcal{R}\left(x_1\right) \subseteq \mathcal{R} \left(r x_0 + (1 - r) x_1\right).$$
 * The  $$\mathcal{R}$$ 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  $$\mathcal{R}$$ is $$\operatorname{Dom} \mathcal{R} := \left\{ x \in X : \mathcal{R}(x) \neq \emptyset \right\}.$$
 * The  $$\mathcal{R}$$ 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 $$\mathcal{S} \circ \mathcal{R} : X \rightrightarrows Z$$ 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) ideally convex.
 * lower cs-closed (lcs-closed) 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:
 * 4) $$B$$ satisfies (H(x, y)) if and only if $$B$$ satisfies (Hx).
 * 5) $$B$$ 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 $$C^i = \operatorname{int} C = \operatorname{int} \left(\operatorname{cl} C\right) = \left(\operatorname{cl} C\right)^i.$$