A Universe of Sorts

Consider the sequence $e_1 = (1, 0, \dots)$ , $e_2 \equiv (0, 1, \dots)$ , and in general $e_i[j] = \delta_i^j$ .
Recall that to check weak convergence, it suffices to check on a basis of the dual space.
We check on the basis $\pi_j (x) \mapsto x[j]$ .
Clearly, on such a basis, we see that $\lim_{n \to \infty} e_n[j] \to 0$ , because after $n > j$ , the sequence will be forever zero.
However, see that this sequence does not strongly converge, since the basis vectors $e_i$ cannot be cauchy, since $||e_i - e_j|| = \sqrt(2)$ when $i \neq j$ .
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.

§ Sequence That Converges Weakly but Not Strongly In lpl^plp.