The question is completely contained in the title :) I can only add, that it is not difficult to give a counterexample for normed spaces, and also Banach-Steinhaus theorem implies the sequential closeness. However, I don't think that this argument can be extended beyond sequences. Thank you.
By the bipolar theorem, this is the case precisely when the topological and algebraic duals coincide. Whether this is true or not for infinite dimensional Banach spaces depends on the set theory you are using.
Edit in response to the comments. Using results of Solovay and Schwartz, the belgian mathematician Garnir showed (in the 70's) that there are set theoretical axiom systems under which every linear functional on a Banach space is continuous.