Finite Dimensional Normed Spaces

Show that if $\mathcal{X}$ is a locally compact normed space, then $\mathcal{X}$ is finite dimensional. (This same result, due to F. Riesz, is valid in the more general topological vector spaces -- see IV.1.1 for the definition. For a nice proof of this look at Pitcairn [1966].)

Let $\mathcal{X}$ be a finite dimensional vector space over $\mathbb {F}$ with Hamel basis $\left\{ e_1, \ldots , e_d\right\}$, and for $x = \sum_{j=1}^{d} x_j e_j$ define $\left\| x\right\|_{\infty } \equiv \max \left\{ \left|x_j\right| : 1 \leq j \leq d\right\}$, as in the proof of Theorem 3.1. Show that $\left\| \cdot \right\|_{\infty }$ is a norm.