I'm trying to prove this identity. Previously, I gave it a try here but the proof there is not correct. I found another proof here which is quite complicated. Could you have a check on my below attempt?
Let $E,F$ be t.v.s. and $E^\star, F^\star$ their continuous dual respectively. Let $\sigma (E, E^\star)$ and $\sigma (F, F^\star)$ be the weak topology of $E, F$ respectively. Let $\sigma (E, E^\star) \boxtimes \sigma (F, F^\star)$ be the product topology of $\sigma (E, E^\star)$ and $\sigma (F, F^\star)$. Then $$\sigma (E, E^\star) \boxtimes \sigma (F, F^\star) = \sigma \big (E \times F, (E \times F)^\star \big).$$
My attempt: By definition, $\sigma (E, E^\star) \boxtimes \sigma (F, F^\star)$ is the coarsest topology on $E \times F$ such that the maps $$ \begin{aligned} \pi_E:E \times F \to E, (x,y) \mapsto x \\ \pi_F:E \times F \to F, (x,y) \mapsto y \end{aligned} $$ are continuous w.r.t. $\sigma (E, E^\star)$ on $E$ and $\sigma (F, F^\star)$ on $F$. Similarly, $\sigma \big (E \times F, (E \times F)^\star \big)$ is the coarsest topology on $E \times F$ such that all the maps $g \in (E \times F)^\star$ are continuous. So it suffices to show that below statements are equivalent.
(S1) $\pi_E,\pi_F$ are continuous.
(S2) $g$ is continuous for all $g \in (E \times F)^\star$.
Let (S1) hold. For each $g \in (E \times F)^\star$, there is $f\in E^\star$ and $h\in F^\star$ such that $g(x,y) = f(x)+h(y)$. Then $g = f \circ \pi_E + h \circ \pi_F$. It follows that $g$ is continuous by S1. Next we need the following lemmas.
Lemma 1: Let $X$ be a set and $(f_i)_{i\in I}$ a family of maps from $X$ to a topological space $Y_i$. We endow $X$ with the initial topology w.r.t. $(f_i)_{i\in I}$. Let $Z$ be a topological space and $\psi:Z \to X$. Then $\psi$ is continuous if and only if $f_i \circ \psi$ is continuous for all $i\in I$.
and
Lemma 2: Let $E$ and $F$ be t.v.s. Then $(E \times F)^\star \cong E^\star \times F^\star$ by the isometric isomorphism $T$ defined by $$T: (E \times F)^\star \to E^\star \times F^\star, g \mapsto (f, h) :=(g\restriction E \times \{0_F\}, g\restriction \{0_E\} \times F).$$ Moreover, $g(x, y) := f(x)+h(y)$.
Let (S2) hold. By symmetry, it's enough to show that $\pi_E$ is continuous. By lemma $1$, it suffices to show that $f \circ \pi_E: (x,y) \mapsto f(x) + 0_{F^\star} (y)$ is continuous for all $f \in E^\star$. By lemma $2$, $f \circ \pi_E \in (E \times F)^\star$. Hence $f \circ \pi_E$ is continuous by S2. This completes the proof.