TITLE: measurable linear functionals are also continuous on separable Banach spaces?
QUESTION [6 upvotes]: It is well known continuous linear functionals are (Borel) measurable. I have read, as a remark, the converse is also true for separable Banach spaces, but I could not find any references.

REPLY [3 votes]: You do not even need seperability. Proposition 1.2.29 in the book Barrelled Locally Convex Spaces of Bonet and Perez Carreras says that every Borel measurable linear map from a Baire locally convex space to a locally convex space is continuous. This result is due to Laurent Schwartz and it is related to his more famous Borelian graph theorem (linear maps from a separable Banach space into Souslin locally convex spaces with borelian graph are continuous).