ca.analysis and odes - Sequence that converge if they have an accumulation point

The following version of the mean ergodic theorem is taken from the book of Krengel, "ergodic theorems".

Let T be a bounded linear operator in a Banach space X. The Birkhoff averages are denoted by $A_n = {1over n} Sigma_{k=0}^{n-1} T^k$.
Assume that the sequence of operator norms $||A_n||$ is bounded independently of $n$. Then for any x and y in B, the following is equivalent :

-- y is a weak cluster point of the sequence $(A_nx)$,

-- y is the weak limit of the sequence $(A_nx)$,

-- y is the strong limit of the sequence $(A_nx)$.

(note that we talk about cluster points instead of converging subsequences because we didn't assume B separable. Hence the weak topology is not necessarily metrizable.)

This theorem implies e.g. the ergodic theorem for Markov operators on $C(K)$ (sequential compactness follows from Azrela-Ascoli), or the ergodic theorem for power bounded operators defined on reflexive Banach spaces (sequential compactness follows from Eberlein-Smulian).

There is a whole set of theorems in ergodic theory along these lines. Let me mention the convergence of the one sided ergodic Hilbert transform, discussed in Cohen and Cuny (see Th 3.2) as another example.