# Are all linear transformations measurable?

Let $V$ and $W$ be topological vector spaces over $\mathbb{F}$ (with $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$), and let $T:V \to W$ be a linear transformation. It is well-known that $T$ is not necessarily continuous. But is $T$ necessarily measurable (with respect to the Borel structures of $V$ and $W$)? Does the answer change when we specifically consider $W=\mathbb{F}$?

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This fails for even finite-dimensional spaces. See math.stackexchange.com/questions/359183/… Or even better yet, see this: artsci.kyushu-u.ac.jp/~ssaito/eng/maths/Cauchy.pdf –  Todd Trimble Aug 1 at 12:06
@ToddTrimble: Thank you for your reply - but in the context of vector spaces, when I say "linear transformation", I mean a transformation that preserves both addition and scalar multiplication. –  Julian Newman Aug 1 at 15:03
Oh, of course you're right -- my apologies. –  Todd Trimble Aug 1 at 15:08

Baire's category theorem (cover the Banach space by the sets where $|f|\leq n$). By the way, it is not customary for questons of this sort to be answered in MO to the last detsil so this is my final word. –  blackburne Aug 1 at 7:05