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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 ).

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.

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

- In this definition, is
**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**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**bcs-complete set**if any Cauchy b-convex series with elements of is convergent and its sum is in

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

If and are topological vector spaces, is a subset of and then is said to satisfy:^{[1]}

**Condition (H**: Whenever is a*x*)*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**Condition (Hw**: Whenever is a*x*)*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 (Hw
*x*).

- If X is locally convex then the statement "and is Cauchy" may be removed from the definition of condition (Hw

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 - 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

- The
**inverse of a multifunction**is the multifunction defined by For any subset - The
**domain of a multifunction**is - The
**image of a multifunction**is For any subset - The composition is defined by for each

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. - (H
*x*) (Hw*x*) 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 (H
*x*) if and only if is cs-closed. - satisfies (Hw
*x*) if and only if is ideally convex.

If is complete then,

- satisfies (H
*x*) if and only if is cs-complete. - satisfies (Hw
*x*) if and only if is bcs-complete. - If and then:
- satisfies (H
*(x, y)*) if and only if satisfies (H*x*). - satisfies (Hw
*(x, y)*) if and only if satisfies (Hw*x*).

- satisfies (H

If is locally convex and is bounded then,

- If satisfies (H
*x*) then is cs-closed. - If satisfies (Hw
*x*) then is ideally convex.

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
- 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
- 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
- Suppose is a Fréchet space and is a subset of If and are lower ideally convex (resp. lower cs-closed) then so is
- Suppose is a Fréchet space and is a multifunction. If are all lower ideally convex (resp. lower cs-closed) then so are and

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

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
- If is ideally convex then