Let $latex X$ be the Banach space $latex C([0,1])$ of continuous functions on the interval $latex [0,1]$ with the sup norm. Consider the following sequence of functions $latex {f_n}$ defind as follows. $latex f(0) = 0 $ and $latex f_n(1/(n+1)) = 1$ for all $latex n = 1, 2, ldots$,  $latex f_n$ is equal to zero on the interval between $latex 2/(n+1)$ and $latex 1$, and $latex f_n$ is linear in the intervals where we haven’t defined it yet (visualize!). The sequence is tending to zero pointwise, but the norm of $latex X$ does not detect this. The sequence tends to $latex 0$ in the $latex L^1$ norm, but the $latex L^1$ norm is not in the game. Can the Banach space structure of $latex X$ detect the convergence of $latex f_n$ to $latex 0$? Continue reading Advanced Analysis, Notes 11: Banach spaces (weak topologies, Alaoglu’s theorem) →