As my friends and I were studying for our real analysis final exam yesterday, we were playing with various examples and found ourselves asking this question:
The space $\ell^1(\mathbb{N})$ is the dual of $c_0(\mathbb{N})$, and the dual of $\ell^1(\mathbb{N})$ is $\ell^\infty(\mathbb{N})$.
Is it possible to have a sequence $\{b_n\}\in\ell^1(\mathbb{N})$ converge to $b\in\ell^1(\mathbb{N})$ weakly*, but not weakly?
We knew that the weak and weak* topologies agree on a reflexive space, and because $\ell^1(\mathbb{N})$ is non-reflexive, we were wondering whether the weak and weak* topologies would agree.
Parsing the definitions, this is just asking whether there is a sequence $\{b_n\}\in\ell^1(\mathbb{N})$ such that, for any $r\in c_0(\mathbb{N})$, we have $\sum_{k=1}^\infty (b_n)_kr_k\to\sum_{k=1}^\infty b_kr_k\quad \text{ as }n\to\infty,$ but for some $s\in \ell^\infty(\mathbb{N})$, $\sum_{k=1}^\infty (b_n)_ks_k\nrightarrow\sum_{k=1}^\infty b_ks_k\quad \text{ as }n\to\infty.$ WLOG, we can let take $b=0$ (just subtract $b$ from all the $b_n$), so this becomes:
Is there a sequence $\{b_n\}\in\ell^1(\mathbb{N})$ such that, for any $r\in c_0(\mathbb{N})$, we have $\sum_{k=1}^\infty (b_n)_kr_k\to 0\quad \text{ as }n\to\infty,$ but for some $s\in \ell^\infty(\mathbb{N})$, $\sum_{k=1}^\infty (b_n)_ks_k\nrightarrow 0\quad \text{ as }n\to\infty ?$
We weren't able to come up with any examples, but of course that doesn't mean there aren't any. Also, just to double-check, were we correct in assuming that it would suffice to check whether weak and weak* convergence of sequences agreed in order to determine whether the weak and weak* topologies agreed?