Convex series

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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 ${\displaystyle \sum _{i=1}^{\infty }r_{i}=1}$).