# Mean ergodic theorem von Neumann

1. Jan 21, 2005

### trosten

I wonder If someone could state the mean ergodic theorem von neumann with out using meassure spaces ? I have studied normed spaces, banach spaces and hilbert spaces, that is complete normed inner product spaces.

Could someone state and explain the theorem for me?

2. Jan 21, 2005

### matt grime

As ergodic theory is the study of invariants measurable spaces, in some loose sense, this doesn't seem very likely. However, reading a statement of it I don't understand what the measure theoretic objection is. One can translate to little l 2 and it states that, modulo some dodgy rearranging of terms, we have approximately:

given an isometry of hilbert space l_2, T, and an element a in l_2, there is a b in l_2 such that

$$\frac{1}{n} \sum_{k=0}^n <T^ka,b>^2$$

tends to zero as n tends to infinity.