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

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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

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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  
  • 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.

Conditions (Hx) and (Hwx)

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

  • 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  
  • 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

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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 multifunction   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  

Relationships

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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,

  1.   is cs-complete (respectively, bcs-complete) if and only if   is cs-closed (respectively, ideally convex).
  2.   satisfies (Hx) if and only if   is cs-closed.
  3.   satisfies (Hwx) if and only if   is ideally convex.

If   is complete then,

  1.   satisfies (Hx) if and only if   is cs-complete.
  2.   satisfies (Hwx) if and only if   is bcs-complete.
  3. If   and   then:
    1.   satisfies (H(x, y)) if and only if   satisfies (Hx).
    2.   satisfies (Hw(x, y)) if and only if   satisfies (Hwx).

If   is locally convex and   is bounded then,

  1. If   satisfies (Hx) then   is cs-closed.
  2. If   satisfies (Hwx) then   is ideally convex.

Preserved properties

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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  

Properties

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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 (resp. lower cs-closed) 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  

See also

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  • Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem

Notes

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  1. ^ Zălinescu 2002, pp. 1–23.

References

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  • 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.