§ Sequence that converges weakly but not strongly in .
- Consider the sequence , , and in general .
- Recall that to check weak convergence, it suffices to check on a basis of the dual space.
- We check on the basis .
- Clearly, on such a basis, we see that , because after , the sequence will be forever zero.
- However, see that this sequence does not strongly converge, since the basis vectors cannot be cauchy, since when .
- The intuition is that weak convergence can only see converge "in a finite subspace", since we are considering what happens with bounded linear functionals.
- Thus, a sequence can appear to converge when restricting attention to any finite region of space, but cannot strongly converge.