Exercise 3, § 2

Let E be a topological vector space over $\mathbf{R}$, and $f$ a homomorphism of the additive group of E in $\mathbf{R}$. Show that if there exists a neighbourhood of 0 in E in which $f$ is bounded, then $f$ is a continuous linear form in E. This is so in particular when $f$ is semi-continuous (lower or upper).