View source for Convex series

{{more footnotes|date=May 2020}}

In mathematics, particularly in [[functional analysis]] and [[convex analysis]], a {{em|{{visible anchor|convex series}}}} is a [[Series (mathematics)|series]] of the form <math>\sum_{i=1}^{\infty} r_i x_i</math> where <math>x_1, x_2, \ldots</math> are all elements of a [[topological vector space]] <math>X</math>, and all <math>r_1, r_2, \ldots</math> are non-negative [[real number]]s that sum to <math>1</math> (that is, such that <math>\sum_{i=1}^{\infty} r_i = 1</math>).

== Types of Convex series ==

Suppose that <math>S</math> is a subset of <math>X</math> and <math>\sum_{i=1}^{\infty} r_i x_i</math> is a convex series in <math>X.</math> 

* If all <math>x_1, x_2, \ldots</math> belong to <math>S</math> then the convex series <math>\sum_{i=1}^{\infty} r_i x_i</math> is called a '''{{visible anchor|convex series}} with elements of <math>S</math>'''.
* If the set <math>\left\{ x_1, x_2, \ldots \right\}</math> is a [[Bounded set (topological vector space)|(von Neumann) bounded set]] then the series called a '''{{visible anchor|b-convex series}}'''. 
* The convex series <math>\sum_{i=1}^{\infty} r_i x_i</math> is said to be a '''{{visible anchor|convergent series}}''' if the sequence of partial sums <math>\left(\sum_{i=1}^n r_i x_i\right)_{n=1}^{\infty}</math> converges in <math>X</math> to some element of <math>X,</math> which is called the '''{{visible anchor|sum of the convex series}}'''. 
* The convex series is called '''{{visible anchor|Cauchy}}''' if <math>\sum_{i=1}^{\infty} r_i x_i</math> is a [[Cauchy series]], which by definition means that the sequence of partial sums <math>\left(\sum_{i=1}^n r_i x_i\right)_{n=1}^{\infty}</math> 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 <math>S</math> is a subset of a [[topological vector space]] <math>X</math> then <math>S</math> is said to be a: 

* '''{{visible anchor|cs-closed set}}''' if any convergent convex series with elements of <math>S</math> has its (each) sum in <math>S.</math> 
** In this definition, <math>X</math> is <em>not</em> required to be Hausdorff, in which case the sum may not be unique. In any such case we require that every sum belong to <math>S.</math>
* '''{{visible anchor|lower cs-closed set}}''' or a '''{{visible anchor|lcs-closed set}}''' if there exists a [[Fréchet space]] <math>Y</math> such that <math>S</math> is equal to the projection onto <math>X</math> (via the canonical projection) of some cs-closed subset <math>B</math> of <math>X \times Y</math> Every cs-closed set is lower cs-closed and every lower cs-closed set is lower ideally convex and [[convex set|convex]] (the converses are not true in general). 
* '''{{visible anchor|ideally convex set}}''' if any convergent b-series with elements of <math>S</math> has its sum in <math>S.</math>
* '''{{visible anchor|lower ideally convex set}}''' or a '''{{visible anchor|li-convex set}}''' if there exists a [[Fréchet space]] <math>Y</math> such that <math>S</math> is equal to the projection onto <math>X</math> (via the canonical projection) of some ideally convex subset <math>B</math> of <math>X \times Y.</math> Every ideally convex set is lower ideally convex. Every lower ideally convex set is convex but the converse is in general not true.
* '''{{visible anchor|cs-complete set}}''' if any Cauchy convex series with elements of <math>S</math> is convergent and its sum is in <math>S.</math>
* '''{{visible anchor|bcs-complete set}}''' if any Cauchy b-convex series with elements of <math>S</math> is convergent and its sum is in <math>S.</math>

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

=== Conditions (Hx) and (Hwx) ===

If <math>X</math> and <math>Y</math> are topological vector spaces, <math>A</math> is a subset of <math>X \times Y,</math> and <math>x \in X</math> then <math>A</math> is said to satisfy:{{sfn|Zălinescu|2002|pp=1-23}} 

* '''{{visible anchor|Condition (H''x'')}}''': Whenever <math>\sum_{i=1}^{\infty} r_i (x_i, y_i)</math> is a {{em|[[#convex series|convex series]]}} with elements of <math>A</math> such that <math>\sum_{i=1}^{\infty} r_i y_i</math> is convergent in <math>Y</math> with sum <math>y</math> and <math>\sum_{i=1}^{\infty} r_i x_i</math> is Cauchy, then <math>\sum_{i=1}^{\infty} r_i x_i</math> is convergent in <math>X</math> and its sum <math>x</math> is such that <math>(x, y) \in A.</math> 
* '''{{visible anchor|Condition (Hw''x'')}}''': Whenever <math>\sum_{i=1}^{\infty} r_i (x_i, y_i)</math> is a {{em|[[#b-convex series|b-convex series]]}} with elements of <math>A</math> such that <math>\sum_{i=1}^{\infty} r_i y_i</math> is convergent in <math>Y</math> with sum <math>y</math> and <math>\sum_{i=1}^{\infty} r_i x_i</math> is Cauchy, then <math>\sum_{i=1}^{\infty} r_i x_i</math> is convergent in <math>X</math> and its sum <math>x</math> is such that <math>(x, y) \in A.</math> 
** If X is locally convex then the statement "and <math>\sum_{i=1}^{\infty} r_i x_i</math> is Cauchy" may be removed from the definition of condition (Hw''x'').

== Multifunctions ==

The following notation and notions are used, where <math>\mathcal{R} : X \rightrightarrows Y</math> and <math>\mathcal{S} : Y \rightrightarrows Z</math> are [[Multivalued function|multifunctions]] and <math>S \subseteq X</math> is a non-empty subset of a [[topological vector space]] <math>X:</math> 

* The '''[[Graph of a multifunction|{{visible anchor|graph of a multifunction}}]] of <math>\mathcal{R}</math>''' is the set <math>\operatorname{gr} \mathcal{R} := \{ (x, y) \in X \times Y : y \in \mathcal{R}(x) \}.</math>
* <math>\mathcal{R}</math> is '''{{visible anchor|closed}}''' (respectively, '''{{visible anchor|cs-closed}}''', '''{{visible anchor|lower cs-closed}}''', '''{{visible anchor|convex}}''', '''{{visible anchor|ideally convex}}''', '''{{visible anchor|lower ideally convex}}''', '''{{visible anchor|cs-complete}}''', '''{{visible anchor|bcs-complete}}''') if the same is true of the graph of <math>\mathcal{R}</math> in <math>X \times Y.</math>
** The multifunction <math>\mathcal{R}</math> is convex if and only if for all <math>x_0, x_1 \in X</math> and all <math>r \in [0, 1],</math> <math>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).</math> 
* The '''{{visible anchor|inverse of a multifunction}} <math>\mathcal{R}</math>''' is the multifunction <math>\mathcal{R}^{-1} : Y \rightrightarrows X</math> defined by <math>\mathcal{R}^{-1}(y) := \left\{ x \in X : y \in \mathcal{R}(x) \right\}.</math> For any subset <math>B \subseteq Y,</math> <math>\mathcal{R}^{-1}(B) := \cup_{y \in B} \mathcal{R}^{-1}(y).</math>
* The '''{{visible anchor|domain of a multifunction}} <math>\mathcal{R}</math>''' is <math>\operatorname{Dom} \mathcal{R} := \left\{ x \in X : \mathcal{R}(x) \neq \emptyset \right\}.</math>
* The '''{{visible anchor|image of a multifunction}} <math>\mathcal{R}</math>''' is <math>\operatorname{Im} \mathcal{R} := \cup_{x \in X} \mathcal{R}(x).</math> For any subset <math>A \subseteq X,</math> <math>\mathcal{R}(A) := \cup_{x \in A} \mathcal{R}(x).</math>
* The {{visible anchor|composition}} <math>\mathcal{S} \circ \mathcal{R} : X \rightrightarrows Z</math> is defined by <math>\left(\mathcal{S} \circ \mathcal{R}\right)(x) := \cup_{y \in \mathcal{R}(x)} \mathcal{S}(y)</math> for each <math>x \in X.</math>

== Relationships ==

Let <math>X, Y, \text{ and } Z</math> be topological vector spaces, <math>S \subseteq X, T \subseteq Y,</math> and <math>A \subseteq X \times Y.</math> The following implications hold:

:complete <math>\implies</math> cs-complete <math>\implies</math> cs-closed <math>\implies</math> lower cs-closed (lcs-closed) {{em|and}} ideally convex.
:lower cs-closed (lcs-closed) {{em|or}} ideally convex <math>\implies</math> lower ideally convex (li-convex) <math>\implies</math> convex.
:(H''x'') <math>\implies</math> (Hw''x'') <math>\implies</math> convex.

The converse implications do not hold in general.

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

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

If <math>X</math> is locally convex and <math>\operatorname{Pr}_X (A)</math> is bounded then,
# If <math>A</math> satisfies (H''x'') then <math>\operatorname{Pr}_X (A)</math> is cs-closed.
# If <math>A</math> satisfies (Hw''x'') then <math>\operatorname{Pr}_X (A)</math> is ideally convex.

=== Preserved properties ===

Let <math>X_0</math> be a linear subspace of <math>X.</math> Let <math>\mathcal{R} : X \rightrightarrows Y</math> and <math>\mathcal{S} : Y \rightrightarrows Z</math> be [[Multivalued function|multifunctions]]. 

* If <math>S</math> is a cs-closed (resp. ideally convex) subset of <math>X</math> then <math>X_0 \cap S</math> is also a cs-closed (resp. ideally convex) subset of <math>X_0.</math> 
* If <math>X</math> is first countable then <math>X_0</math> is cs-closed (resp. cs-complete) if and only if <math>X_0</math> is closed (resp. complete); moreover, if <math>X</math> is locally convex then <math>X_0</math> is closed if and only if <math>X_0</math> is ideally convex. 
* <math>S \times T</math> is cs-closed (resp. cs-complete, ideally convex, bcs-complete) in <math>X \times Y</math> if and only if the same is true of both <math>S</math> in <math>X</math> and of <math>T</math> in <math>Y.</math>
* 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 <math>X</math> 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 <math>X</math> 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 <math>X</math> is a [[Fréchet space]] and the <math>A</math> and <math>B</math> are subsets. If <math>A</math> and <math>B</math> are lower ideally convex (resp. lower cs-closed) then so is <math>A + B.</math>
* Suppose <math>X</math> is a [[Fréchet space]] and <math>A</math> is a subset of <math>X.</math> If <math>A</math> and <math>\mathcal{R} : X \rightrightarrows Y</math> are lower ideally convex (resp. lower cs-closed) then so is <math>\mathcal{R}(A).</math>
* Suppose <math>Y</math> is a [[Fréchet space]] and <math>\mathcal{R}_2 : X \rightrightarrows Y</math> is a multifunction. If <math>\mathcal{R}, \mathcal{R}_2, \mathcal{S}</math> are all lower ideally convex (resp. lower cs-closed) then so are <math>\mathcal{R} + \mathcal{R}_2 : X \rightrightarrows Y</math> and <math>\mathcal{S} \circ \mathcal{R} : X \rightrightarrows Z.</math>

== Properties ==

If <math>S</math> be a non-empty convex subset of a topological vector space <math>X</math> then,
# If <math>S</math> is closed or open then <math>S</math> is cs-closed.
# If <math>X</math> is [[Hausdorff space|Hausdorff]] and finite dimensional then <math>S</math> is cs-closed.
# If <math>X</math> is [[first countable]] and <math>S</math> is ideally convex then <math>\operatorname{int} S = \operatorname{int} \left(\operatorname{cl} S\right).</math>

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

If <math>X</math> is a barreled [[first countable]] space and if <math>C \subseteq X</math> then:
# If <math>C</math> is lower ideally convex then <math>C^i = \operatorname{int} C,</math> where <math>C^i := \operatorname{aint}_X C</math> denotes the [[algebraic interior]] of <math>C</math> in <math>X.</math>
# If <math>C</math> is ideally convex then <math>C^i = \operatorname{int} C = \operatorname{int} \left(\operatorname{cl} C\right) = \left(\operatorname{cl} C\right)^i.</math>

== See also ==

* {{annotated link|Ursescu theorem}}

== Notes ==

{{reflist|group=note}}
{{reflist}}

== References ==

* {{Zălinescu Convex Analysis in General Vector Spaces 2002}} <!-- {{sfn|Zălinescu|2002|pp=}} -->
* {{Cite journal|last=Baggs|first=Ivan|date=1974|title=Functions with a closed graph|url=https://www.ams.org/|journal=Proceedings of the American Mathematical Society|language=en|volume=43|issue=2|pages=439–442|doi=10.1090/S0002-9939-1974-0334132-8|issn=0002-9939|doi-access=free}}

{{Functional analysis}}
{{Convex analysis and variational analysis}}
{{Analysis in topological vector spaces}}

[[Category:Theorems in functional analysis]]