From Euclidean to Hilbert Spaces

Appendix 3Uniform, Strong and Weak Convergence

Sequences of operators may be shown to converge with respect to different topologies than the one induced by the operator norm. The same can be said for sequences of elements in Banach or Hilbert spaces.

To take a concrete example, consider the following case. Let (u_n)_n∈ℕ be an arbitrary Hilbert basis in a Hilbert space . For all n ∈ ℕ, we define the linear operator:

From the geometric characterization of projection operators (see Theorem 6.32), we know that A_n is the orthogonal projector on the vector subspace of generated by u₁, . . . , u_n: S_n = span(u₁, . . . , u_n).

Since any x ∈ may be written as , it would seem that the sequence of projectors (A_n)_n∈ℕ converges toward id when n → +∞.

Nevertheless, since S_n ⊂ S_n′ ∀n < n^′, we know by Theorem 6.35 that A_n′ – A_n is the projector onto , thanks to the orthogonality of the vectors (u_n)_n∈ℕ