According to Banach and Saks, every bounded sequence in or () has a subsequence whose Cesaro-means converge strongly. More generally, every uniformly convex Banach space possesses this so-called Banach-Saks property, as shown by Kakutani. In particular, every Hilbert space has this property. In nonlinear analysis, by utilizing a duality mapping some assertions which are valid in the case of Hilbert spaces are extended to the case of special classes of Banach spaces. Especially in the case of Banach spaces with a uniformly convex conjugate space, such extentions are often obtained since a duality mapping is uniformly strongly continuous on each bounded subset of such a Banach space.

The Banach-Saks theorem in states that

Theorem (Banach-Saks for spaces). Given in a sequence which converges weakly to an element , we can select a subsequence such that the arithmetic meansconverge in strongly to .

This theorem is due to the two Polish geometers S. Banach and S. Saks, whose work and, in particular, the importance of whose research in the topics treated in this book are widely acknowledged.

*Proof*. Replacing by , we can assume . We shall successively choose the in the following manner. Beginning for definiteness with , let be an index or, if we wish, the first index such that

;

this choice is possible since as . In general, after having chosen we choose so that

,

which is possible since as and . Since, furthermore, the norms form a bounded sequence, say , it follows by expanding the inner product that

which implies

as .

Corollary. If a linear set, or more generally a convex set, in is closed in the sense of weak convergence, it is also closed in the sense of strong convergence.

In a Hilbert space, we still have the following version:

Theorem (Banach-Saks for Hilbert spaces). Every bounded sequence contains a subsequence and a point such thatconverges strongly to as goes to infinity.

In a Banach space, we have the following version:

Theorem (Banach-Saks for Banach spaces). If a sequence of elements converges weakly to an element , then there exists some subsequence of elements such that the averagesconverges strongly to as goes to infinity.

It is natural enough to ask whether the same property is true for functionals that converge in the weak-* topology, but unfortunately (as is well known) the answer is no. Consider the sequence of functions

in to be a sequence of linear functionals on . This sequence converges in the weak-* topology to by the Riemann-Lebesgue lemma, yet for any subsequence, the average will have a norm of since each function approaches as tends to .

Cf: S. Banach and S. Saks: Sur la convergence forte dans les champs , * Studia Math.* **2** (1930), pp. 51-57.

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