[proofplan] We verify additivity and homogeneity of the evaluation map $\mathrm{ev}: V \to V^{**}$ directly from the linearity of functionals in $V^*$. [/proofplan] [step:Verify additivity of the evaluation map] For $u, v \in V$ and any $\theta \in V^*$: \begin{align*} \mathrm{ev}(u + v)(\theta) = \theta(u + v) = \theta(u) + \theta(v) = \mathrm{ev}(u)(\theta) + \mathrm{ev}(v)(\theta). \end{align*} Since this holds for all $\theta \in V^*$: $\mathrm{ev}(u + v) = \mathrm{ev}(u) + \mathrm{ev}(v)$. [/step] [step:Verify homogeneity of the evaluation map] For $c \in \mathbb{F}$, $v \in V$, and any $\theta \in V^*$: \begin{align*} \mathrm{ev}(cv)(\theta) = \theta(cv) = c\,\theta(v) = c\,\mathrm{ev}(v)(\theta). \end{align*} Since this holds for all $\theta \in V^*$: $\mathrm{ev}(cv) = c\,\mathrm{ev}(v)$. [/step]