A sequence $(x_n)$ in a normed linear space $X$ is said to converge weakly to $x$ if $ \lim _{n\rightarrow\infty} \ell(x_n) = \ell(x)$ Consider the sequence $(f_n) \in C([0,1])$ defined by $f_n(t) = t^n.$ Does this sequence converge weakly? Does it have weakly convergent subsequences?
I don't really see the sequences here, do we get one different sequence depending on what $t$ is? $ \lim _{n\rightarrow\infty} \ell f_n(t) = 0, $ for $t \neq 1$. So it seems that f would converge to something not continuous?