§ Sequence that converges weakly but not strongly in lp.
Consider the sequence e1=(1,0,…), e2≡(0,1,…), and in general ei[j]=δij.
Recall that to check weak convergence, it suffices to check on a basis of the dual space.
We check on the basis πj(x)↦x[j].
Clearly, on such a basis, we see that limn→∞en[j]→0, because after n>j, the sequence will be forever zero.
However, see that this sequence does not strongly converge, since the basis vectors ei cannot be cauchy, since ∣∣ei−ej∣∣=(2) when i=j.
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.