I'm attempting to answer the following:
Let X be a reflexive Banach space, and Y be a normed space. Suppose T : X → Y is a linear operator with the property that if a sequence $x_n$ converges weakly to 0 in X, then $T(x_n)$ converges weakly to 0 in Y . Prove that T is bounded.
I feel like I can use the Banach-Alaoglu Theorem to get there, namely that it guarantees that every bounded sequence in X has a weakly convergent subsequence, but I'm not sure how to proceed.
Any help is much appreciated!