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 $W=\mathbb{F}$?



Are all linear transformations measurable?
This question is answered by the well-known construction of a non-continuous linear form on an infinite dimensional Banach space using Hamel bases. Note also that there is a measurable graph theorem (L. Schwartz) which implies that all measurable linear maps, say between separable Banach spaces, are continuous. And there are models of set theory in which ALL linear mappings between suitable classes of spaces are continuous (Solovay and Garnir).


