Normed Linear Space - 4. The Dual Space
4. The Dual Space For two normed linear spaces $V,\, W$ over $\F$, define two spaces \[ \begin{aligned} \mathcal{L}(V,\,W) &:= \set{L:V\to W}{\text{$L$ is linear and continuous}} \\ \mathcal{B}(V,\,W) &:= \set{L:V\to W}{\text{$L$ is linear and bounded}} \\ \end{aligned} \] We have seen that $\mathcal{L}(V,\,W) = \mathcal{B}(V,\,W)$. For $L,\, L_1,\, L_2 \in \mathcal{L}(V,\,W)$, define \[ \begin{..