Remark 2

Suppose $K = \mathbf{R}$. We leave it to the reader to verify that formulas (40) and (41) are again valid, and to show that $\mathscr{L}^2(E; F)$ can be identified with the dual of $\mathscr{L}^2(F; E)$ by means of the bilinear form $(u, v) \mapsto \operatorname{Tr}(uv)$.