§ Sequence that converges weakly but not strongly in $l^p$.

• 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.