Convex series


In mathematics, particularly in functional analysis and convex analysis, a 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 .

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 with elements of .
  • If the set is a (von Neumann) bounded set then the series called a '.
  • The convex series is said to be a ' if the sequence of partial sums converges in to some element of which is called the '.
  • The convex series is called ' 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: ' if any convergent convex series with elements of has its 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
  • ' or a ' if there exists a Fréchet space such that is equal to the projection onto 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.
  • ' if any convergent b-series with elements of has its sum in ' or a ' if there exists a Fréchet space such that is equal to the projection onto 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.' if any Cauchy convex series with elements of is convergent and its sum is in
  • ' 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.

Conditions (Hx) and (Hwx)

If and are topological vector spaces, is a subset of and then is said to satisfy: ': Whenever is a with elements of such that is convergent in with sum and is Cauchy, then is convergent in and its sum is such that
  • ': Whenever is a 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.

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
  • is if the same is true of the graph of in
  • * The multifunction is convex if and only if for all and all
  • The is the multifunction defined by For any subset
  • The is
  • The is For any subset
  • The is defined by for each

Relationships

Let be topological vector spaces, and The following implications hold:
The converse implications do not hold in general.
If is complete then,
  1. is cs-complete if and only if is cs-closed.
  2. satisfies if and only if is cs-closed.
  3. satisfies if and only if is ideally convex.
If is complete then,
  1. satisfies if and only if is cs-complete.
  2. satisfies if and only if is bcs-complete.
  3. If and then:
  4. # satisfies if and only if satisfies.
  5. # satisfies if and only if satisfies.
If is locally convex and is bounded then,
  1. If satisfies then is cs-closed.
  2. If satisfies then is ideally convex.

Preserved properties

Let be a linear subspace of Let and be multifunctions.
  • If is a cs-closed subset of then is also a cs-closed subset of
  • If is first countable then is cs-closed if and only if is closed ; moreover, if is locally convex then is closed if and only if is ideally convex.
  • is cs-closed 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 subsets of has the same property.
  • The Cartesian product of cs-closed subsets of arbitrarily many topological vector spaces has that same property.
  • The intersection of countably many lower ideally convex subsets of has the same property.
  • The Cartesian product of lower ideally convex subsets of countably many topological vector spaces has that same property.
  • Suppose is a Fréchet space and the and are subsets. If and are lower ideally convex then so is
  • Suppose is a Fréchet space and is a subset of If and are lower ideally convex then so is
  • Suppose is a Fréchet space and is a multifunction. If are all lower ideally convex then so are and

Properties

If be a non-empty convex subset of a topological vector space then,
  1. If is closed or open then is cs-closed.
  2. If is Hausdorff and finite dimensional then is cs-closed.
  3. 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 then the same is true of
If is a barreled first countable space and if then:
  1. If is lower ideally convex then where denotes the algebraic interior of in
  2. If is ideally convex then