After proving that the weak topology is compatible with the vector space structure, I try to do the same for weak$^\star$ topology, i.e.,
Let $E$ be a topological vector space and $E^\star$ its topological dual. Let $\sigma(E^\star, E)$ be the weak$^\star$ topology on $E^\star$. We denote by $E^\star_{w^\star}$ the vector space $E^\star$ together with $\sigma(E^\star, E)$. Then $E^\star_{w^\star}$ is a topological vector space.
I posted my proof as an answer below. Could you have a check on my attempt?
Consider the canonical injection $J:E \to E^{\star\star}, x \mapsto (f \mapsto \langle f, x \rangle)$. Then $\sigma(E^\star, E)$ is the initial topology on $E^\star$ w.r.t. the collection $(Jx)_{x\in E}$ of maps. We consider the linear maps $$\begin{aligned} T&:E^\star_{w^\star} \times E^\star_{w^\star} &&\to E^\star_{w^\star}, (f, h) &&\mapsto f+h \\ L&: \mathbb R \times E^\star_{w^\star} &&\to E^\star_{w^\star}, (t, f) &&\mapsto tf \end{aligned}.$$
We want to prove that $T,L$ are continuous. It suffices to show that $$\begin{aligned} &Jx \circ T:E^\star_{w^\star} \times E^\star_{w^\star} &&\to \mathbb R, (f, h) &&\mapsto \langle Jx, f \rangle + \langle Jx, h \rangle \\ &Jx \circ L: \mathbb R \times E^\star_{w^\star} &&\to \mathbb R, (t, f) &&\mapsto t\langle Jx, f \rangle \end{aligned}$$ are continuous for all $x \in E$. The claim then follows from below diagrams $$ \begin{aligned} &\substack{E^\star_{w^\star} \times E^\star_{w^\star} \\ (f, h)} &&\substack{ \longrightarrow \\ \longmapsto } \, \substack{\mathbb R \times \mathbb R \\ ( \langle Jx, f \rangle, \langle Jx, h \rangle)} && \substack{ \longrightarrow \\ \longmapsto } \, \substack{ \mathbb R \\ \langle Jx, f \rangle + \langle Jx, h \rangle} \\ &\substack{\mathbb R \times E^\star_{w^\star} \\ (t, f)} &&\substack{ \longrightarrow \\ \longmapsto } \, \substack{\mathbb R \times \mathbb R \\ (t, \langle Jx, f \rangle)} && \substack{ \longrightarrow \\ \longmapsto } \, \substack{ \mathbb R \\ t\langle Jx, f \rangle} \end{aligned} $$ and the fact that $Jx$ is also continuous in the weak$^\star$ topology.