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In mathematics, particularly in functional analysis and convex analysis, a convex series is a series of the form ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$ where ${\displaystyle x_{1},x_{2},\ldots }$ are all elements of a topological vector space ${\displaystyle X}$, and all ${\displaystyle r_{1},r_{2},\ldots }$ are non-negative real numbers that sum to ${\displaystyle 1}$ (that is, such that ${\displaystyle \sum _{i=1}^{\infty }r_{i}=1}$).

Types of Convex series

Suppose that ${\displaystyle S}$ is a subset of ${\displaystyle X}$ and ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$ is a convex series in ${\displaystyle X.}$

• If all ${\displaystyle x_{1},x_{2},\ldots }$ belong to ${\displaystyle S}$ then the convex series ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$ is called a convex series with elements of ${\displaystyle S}$.
• If the set ${\displaystyle \left\{x_{1},x_{2},\ldots \right\}}$ is a (von Neumann) bounded set then the series called a b-convex series.
• The convex series ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$ is said to be a convergent series if the sequence of partial sums ${\displaystyle \left(\sum _{i=1}^{n}r_{i}x_{i}\right)_{n=1}^{\infty }}$ converges in ${\displaystyle X}$ to some element of ${\displaystyle X,}$ which is called the sum of the convex series.
• The convex series is called Cauchy if ${\displaystyle \sum _{i=1}^{\infty }r_{i}x_{i}}$ is a Cauchy series, which by definition means that the sequence of partial sums ${\displaystyle \left(\sum _{i=1}^{n}r_{i}x_{i}\right)_{n=1}^{\infty }}$ 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 ${\displaystyle S}$ is a subset of a topological vector space ${\displaystyle X}$ then ${\displaystyle S}$ is said to be a:

• cs-closed set if any convergent convex series with elements of ${\displaystyle S}$ has its (each) sum in ${\displaystyle S.}$
  • In this definition, ${\displaystyle X}$ 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 ${\displaystyle S.}$
• lower cs-closed set or a lcs-closed set if there exists a Fréchet space ${\displaystyle Y}$ such that ${\displaystyle S}$ is equal to the projection onto ${\displaystyle X}$ (via the canonical projection) of some cs-closed subset ${\displaystyle B}$ of ${\displaystyle X\times Y}$ 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 ${\displaystyle S}$ has its sum in ${\displaystyle S.}$
• lower ideally convex set or a li-convex set if there exists a Fréchet space ${\displaystyle Y}$ such that ${\displaystyle S}$ is equal to the projection onto ${\displaystyle X}$ (via the canonical projection) of some ideally convex subset ${\displaystyle B}$ of ${\displaystyle X\times Y.}$ 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 ${\displaystyle S}$ is convergent and its sum is in ${\displaystyle S.}$
• bcs-complete set if any Cauchy b-convex series with elements of ${\displaystyle S}$ is convergent and its sum is in ${\displaystyle S.}$

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

Conditions (Hx) and (Hwx)

If ${\displaystyle X}$ and ${\displaystyle Y}$ are topological vector spaces, ${\displaystyle A}$ is a subset of ${\displaystyle X\times Y,}$ and ${\displaystyle x\in <!--\; \in \; -->X}$Zdroj:https://en.wikipedia.org?pojem=Convex_series
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