**Forward direction.** Suppose $f_\alpha \to f$ in $\sigma(E', E)$. Let $x \in E$ be arbitrary. By definition of $\sigma(E', E)$, the evaluation map \begin{align*} \operatorname{ev}_x: E' &\to \mathbb{R} \\ g &\mapsto g(x) \end{align*} is [continuous](/page/Continuity). Composing a continuous map with a convergent net preserves convergence: \begin{align*} f_\alpha(x) = \operatorname{ev}_x(f_\alpha) \to \operatorname{ev}_x(f) = f(x). \end{align*} Since $x \in E$ was arbitrary, $f_\alpha(x) \to f(x)$ for every $x \in E$. **Reverse direction.** Suppose $f_\alpha(x) \to f(x)$ for every $x \in E$. Let $W$ be an arbitrary $\sigma(E', E)$-neighbourhood of $f$. By the definition of the initial [topology](/page/Topology) generated by $\{\operatorname{ev}_x\}_{x \in E}$, the [set](/page/Set) $W$ contains a basic neighbourhood of $f$: there exist finitely many vectors $x_1, \ldots, x_m \in E$ and a tolerance $\varepsilon > 0$ such that \begin{align*} V(f;\, x_1, \ldots, x_m;\, \varepsilon) &:= \{g \in E' : |g(x_i) - f(x_i)| < \varepsilon \text{ for all } i = 1, \ldots, m\} \subseteq W. \end{align*} [claim: Finitely Many Pointwise Conditions Eventually Hold Simultaneously] There exists $\alpha_0 \in A$ such that for every $\alpha \geq \alpha_0$ and every $i \in \{1, \ldots, m\}$, \begin{align*} |f_\alpha(x_i) - f(x_i)| < \varepsilon. \end{align*} [/claim] [proof] By hypothesis, for each $i \in \{1, \ldots, m\}$, the net $(f_\alpha(x_i))_{\alpha \in A}$ converges to $f(x_i)$ in $\mathbb{R}$. So for each $i$ there exists $\alpha_i \in A$ such that \begin{align*} |f_\alpha(x_i) - f(x_i)| < \varepsilon \quad \text{for all } \alpha \geq \alpha_i. \end{align*} Since $A$ is a directed set and $\{1, \ldots, m\}$ is finite, there exists $\alpha_0 \in A$ satisfying $\alpha_0 \geq \alpha_i$ for all $i = 1, \ldots, m$. For every $\alpha \geq \alpha_0$, we have $\alpha \geq \alpha_i$ for each $i$, so the bound holds for all $i$ simultaneously. [/proof] For every $\alpha \geq \alpha_0$, the claim gives \begin{align*} f_\alpha \in V(f;\, x_1, \ldots, x_m;\, \varepsilon) \subseteq W. \end{align*} Since $W$ was an arbitrary $\sigma(E', E)$-neighbourhood of $f$, the net $f_\alpha$ is eventually in every neighbourhood of $f$, so $f_\alpha \to f$ in $\sigma(E', E)$.