Banach-Saks property
Banach-Saks property is a property of certain normed vector spaces stating that every bounded sequence of points in the space has a subsequence that is convergent in the mean (also known as Cesàro summation or limesable). Specifically, for every bounded sequence ( x n ) n {\displaystyle (x_{n})_{n}} in the space, there exists a subsequence ( x n k ) k {\displaystyle (x_{n_{k}})_{k}} such that the sequence ( x n 1 + … + x n k k ) k = 1 ∞ {\displaystyle \left({\frac {x_{n_{1}}+\ldots +x_{n_{k}}}{k}}\right)_{k=1}^{\infty }} is convergent (in the sense of the norm).