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