Proposition (pointwise limit of continuous linear functions from a barrelled LCTVS into a Hausdorff TVS is continuous and linear):
Let be a barrelled LCTVS over a field , let be a locally closed Hausdorff TVS over the same field, and suppose that a net () of linear and continuous functions is given. Suppose further that is a function such that
Then is itself a linear and continuous functional.
Topological tensor products
Tensor product of Hilbert spacesEdit
Proposition (tensor product of orthonormal bases is orthonormal basis of tensor product):
Let be Hilbert spaces, and suppose that is an orthonormal basis of and is an orthonormal basis of . Then is an orthonormal basis of .
Proof: Let any element
of be given; by definition, each element of may be approximated by such elements. Let . Then by definition of an orthonormal basis, we find for and for and then resp. such that
- and .
Then note that by the triangle inequality,
Now fix . Then by the triangle inequality,
In total, we obtain that
(assuming that the given sum approximates well enough) which is arbitrarily small, so that the span of tensors of the form is dense in .
Now we claim that the basis is orthonormal. Indeed, suppose that . Then
Similarly, the above expression evaluates to when and . Hence, does constitute an orthonormal basis of .
Theorem (Von Neumann ergodic theorem):
Let be Hilbert space, and let be a unitary operator. Further, let the orthogonal projection onto the space be given by . Then
where the limit is taken with respect to the operator norm on , the space of bounded operators on . Moreover, the inequality
is a valid estimate for the convergence rate.
Proof: Suppose first that and . Then
Further, if we set
If now the sequence is convergent, we see that its limit is indeed contained within . From the respective former consideration, we may hence infer that the sequence does in fact converge to . We are thus reduced to proving the convergence of the sequence in operator norm. Since is Hilbert space, proving that is a Cauchy sequence will be sufficient. But since
for this is the case; the gaps are closed using that
Taking in the next to last computation yields the desired rate of convergence. These computations also reveal the underlying cause of convergence: The sequence becomes more and more uniform, since applying to it does not change it by a large amount.