Banach-Saks Theorem -- from Wolfram MathWorld

The Banach-Saks theorem is a result in functional analysis which proves the existence of a "nicely-convergent" subsequence for any sequence ${f_n}={f_n}_(n in Z^*)$ of functions provided that the elements of the sequence possess certain other convergence and integrability properties. The theorem is sometimes referred to as the Banach-Saks-Mazur theorem due to Mazur's generalizations of the original result.

To state the result precisely, let be a real number satisfying , let be a "suitably nice" measure on (e.g., the standard Lebesgue measure ), and let ${f_n}$ be a sequence of functions in which converges weakly to a function . The Banach-Saks theorem states that the sequence ${f_n}$ necessarily has a subsequence ${f_(n_k)}$ for which the so-called Cesàro mean

converges in mean to as tends to infinity.

The above-stated version of the Banach-Saks theorem has a number of useful corollaries which are utilized ubiquitously throughout functional analysis. For example, this result implies that a convex set of functions in space which is closed with respect to convergence in mean is necessarily closed in the sense of weak convergence.

It should also be noted that the above can be adapted and rephrased for a higher degree of generality. For example, the above-stated version of the theorem remains true if is replaced by and if weak convergence of the sequence ${f_n}$ is replaced by boundedness. The result also holds for sequences taken from a number of spaces other than , e.g., for sequences of functions in the space consisting of all continuous real-valued functions with continuous first derivatives and for arbitrary bounded sequences in uniformly convex Banach spaces. One case of the Banach-Saks theorem which is of particular interest in functional analysis is stated in the language of Hilbert spaces and says that every bounded sequence ${x_n}={x_n}_(n in Z^*)$ in a Hilbert space contains a subsequence ${x_(n_k)}$ whose Cesàro means converge strongly to some point . These results have been further generalized to so-called -Hilbert spaces and to Banach spaces whose conjugate space (that is, the complex conjugate of the dual vector space ) are uniformly convex.