In mathematics, particularly in functional analysis and convex analysis, a convex series is a series of the form
where
are all elements of a topological vector space
, and all
are non-negative real numbers that sum to
(that is, such that
).
Types of Convex series
Suppose that
is a subset of
and
is a convex series in
- If all
belong to
then the convex series
is called a convex series with elements of
. - If the set
is a (von Neumann) bounded set then the series called a b-convex series. - The convex series
is said to be a convergent series if the sequence of partial sums
converges in
to some element of
which is called the sum of the convex series. - The convex series is called Cauchy if
is a Cauchy series, which by definition means that the sequence of partial sums
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
is a subset of a topological vector space
then
is said to be a:
- cs-closed set if any convergent convex series with elements of
has its (each) sum in
- In this definition,
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23bbb1f0f6ebdfa78b4fed06049640f7386bb44b)
- lower cs-closed set or a lcs-closed set if there exists a Fréchet space
such that
is equal to the projection onto
(via the canonical projection) of some cs-closed subset
of
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
has its sum in ![{\displaystyle S.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23bbb1f0f6ebdfa78b4fed06049640f7386bb44b)
- lower ideally convex set or a li-convex set if there exists a Fréchet space
such that
is equal to the projection onto
(via the canonical projection) of some ideally convex subset
of
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
is convergent and its sum is in ![{\displaystyle S.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23bbb1f0f6ebdfa78b4fed06049640f7386bb44b)
- bcs-complete set if any Cauchy b-convex series with elements of
is convergent and its sum is in ![{\displaystyle S.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23bbb1f0f6ebdfa78b4fed06049640f7386bb44b)
The empty set is convex, ideally convex, bcs-complete, cs-complete, and cs-closed.
Conditions (Hx) and (Hwx)
If
and
are topological vector spaces,
is a subset of
and
then
is said to satisfy:
- Condition (Hx): Whenever
is a convex series with elements of
such that
is convergent in
with sum
and
is Cauchy, then
is convergent in
and its sum
is such that ![{\displaystyle (x,y)\in A.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/748df670a00e39ff1f37cf6df674d10d55439916)
- Condition (Hwx): Whenever
is a b-convex series with elements of
such that
is convergent in
with sum
and
is Cauchy, then
is convergent in
and its sum
is such that
- If X is locally convex then the statement "and
is Cauchy" may be removed from the definition of condition (Hwx).
Multifunctions
The following notation and notions are used, where
and
are multifunctions and
is a non-empty subset of a topological vector space
- The graph of a multifunction of
is the set ![{\displaystyle \operatorname {gr} {\mathcal {R}}:=\{(x,y)\in X\times Y:y\in {\mathcal {R}}(x)\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c267030116c33431d13f7034946b5768d2767362)
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
in
- The mulifunction
is convex if and only if for all
and all
![{\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5b087724afbc62d97cac086fbc15bb23fbdd95)
- The inverse of a multifunction
is the multifunction
defined by
For any subset
![{\displaystyle {\mathcal {R}}^{-1}(B):=\cup _{y\in B}{\mathcal {R}}^{-1}(y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbfae1aa0e0590057a4873f1022da9fd89db5a0c)
- The domain of a multifunction
is ![{\displaystyle \operatorname {Dom} {\mathcal {R}}:=\left\{x\in X:{\mathcal {R}}(x)\neq \emptyset \right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2d012c9299af09bbd0316eb4bc204070123d36)
- The image of a multifunction
is
For any subset
![{\displaystyle {\mathcal {R}}(A):=\cup _{x\in A}{\mathcal {R}}(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1561584d48fe547faa6c9ea070c74acfbb8d7915)
- The composition
is defined by
for each ![{\displaystyle x\in X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0deab6a01578b5b543b772df12dc0d2c593cc924)
Relationships
Let
be topological vector spaces,
and
The following implications hold:
- complete
cs-complete
cs-closed
lower cs-closed (lcs-closed) and ideally convex. - lower cs-closed (lcs-closed) or ideally convex
lower ideally convex (li-convex)
convex. - (Hx)
(Hwx)
convex.
The converse implications do not hold in general.
If
is complete then,
is cs-complete (respectively, bcs-complete) if and only if
is cs-closed (respectively, ideally convex).
satisfies (Hx) if and only if
is cs-closed.
satisfies (Hwx) if and only if
is ideally convex.
If
is complete then,
satisfies (Hx) if and only if
is cs-complete.
satisfies (Hwx) if and only if
is bcs-complete. - If
and
then:
satisfies (H(x, y)) if and only if
satisfies (Hx).
satisfies (Hw(x, y)) if and only if
satisfies (Hwx).
If
is locally convex and
is bounded then,
- If
satisfies (Hx) then
is cs-closed. - If
satisfies (Hwx) then
is ideally convex.
Preserved properties
Let
be a linear subspace of
Let
and
be multifunctions.
- If
is a cs-closed (resp. ideally convex) subset of
then
is also a cs-closed (resp. ideally convex) subset of ![{\displaystyle X_{0}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55b7c923377f1df557894dcca213d88956b0c266)
- If
is first countable then
is cs-closed (resp. cs-complete) if and only if
is closed (resp. complete); moreover, if
is locally convex then
is closed if and only if
is ideally convex.
is cs-closed (resp. cs-complete, ideally convex, bcs-complete) in
if and only if the same is true of both
in
and of
in ![{\displaystyle Y.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61)
- 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
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
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
is a Fréchet space and the
and
are subsets. If
and
are lower ideally convex (resp. lower cs-closed) then so is ![{\displaystyle A+B.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb5ee62a8753fecbb578e3703fc38062b193b7d8)
- Suppose
is a Fréchet space and
is a subset of
If
and
are lower ideally convex (resp. lower cs-closed) then so is ![{\displaystyle {\mathcal {R}}(A).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff34611493c2bbbd824c0d09b6602d8bc7067eb0)
- Suppose
is a Fréchet space and
is a multifunction. If
are all lower ideally convex (resp. lower cs-closed) then so are
and ![{\displaystyle {\mathcal {S}}\circ {\mathcal {R}}:X\rightrightarrows Z.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ca185841eff5a7dd2ca3202b58511587bfb036)
Properties
If
be a non-empty convex subset of a topological vector space
then,
- If
is closed or open then
is cs-closed. - If
is Hausdorff and finite dimensional then
is cs-closed. - If
is first countable and
is ideally convex then ![{\displaystyle \operatorname {int} S=\operatorname {int} \left(\operatorname {cl} S\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b4f38b003d100f45f10570348a34a1109985724)
Let
be a Fréchet space,
be a topological vector spaces,
and
be the canonical projection. If
is lower ideally convex (resp. lower cs-closed) then the same is true of
If
is a barreled first countable space and if
then:
- If
is lower ideally convex then
where
denotes the algebraic interior of
in ![{\displaystyle X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03)
- If
is ideally convex then ![{\displaystyle C^{i}=\operatorname {int} C=\operatorname {int} \left(\operatorname {cl} C\right)=\left(\operatorname {cl} C\right)^{i}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81ed9036d21ea6a023ef41372377a48dbafe43d5)
See also
- Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
Notes
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.
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