The Banach-Saks theorem is a result in functional analysis which proves the existence of a "nicely-convergent" subsequence
for any sequence 
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
(Soenjaya 2013).
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 
be a sequence of functions
in 
which converges weakly to a function 
. The Banach-Saks theorem states
that the sequence 
necessarily has a subsequence 
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 Lp-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 
is replaced by boundedness.
The result also holds for arbitrary bounded sequences
in uniformly convex Banach
spaces. It does not hold for arbitrary bounded sequences in ![C^1([0,1])](/images/equations/Banach-SaksTheorem/Inline17.svg)
with its usual norm. 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 
in a Hilbert
space 
contains a subsequence 
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 (an older name
for the continuous dual 
) is uniformly convex.
Spaces whose sequences satisfy a version of the Banach-Saks theorem are sometimes said to possess the Banach-Saks property.
-Hilbert Spaces." J. Indones. Math. Soc. 19,
79-87, 2013.