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.

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.