Weak* convergence in terms of weak* topology

Question

I want to ask a seemingly easy question, but I have no idea how to show it. Let $X$ be a locally convex space over $F$. $\{T_n\}$ be a sequence in $X^\star$ and $T\in X^\star$. I know $\{T_n\}$ is called weak $\star$ convergent to $T$ if $\lim_{n\to\infty}T_n(x)=T(x)$ for all $x\in X$. On the other hand, from $X$, we can construct a weak $\star$ topology of $X^\star$ by seminorms $\{x^\star\mapsto x^\star(x)\mid x\in X\}$. However, how to prove that $\{T_n\}$ is weak $\star$ convergent to $T$ iff $\{T_n\}$ is convergent (as a sequence in the space $X^\star$) with respect to the weak $\star$ topology?

Answer 1

You need a description of a system of basic weak-star neighborhoods of a point $\phi\in X^*$.

Such a description is given by the sets $$V_\phi(\varepsilon, F):=\{\psi\in X^*: |\psi(x)-\phi(x)|<\varepsilon\text{ for all }x\in F\}$$ where $\varepsilon$ ranges in $(0,\infty)$ and $F$ ranges in the finite subsets of $X$. It is elementary to verify that the collection $\{V_\phi(\varepsilon,F)\}_{\varepsilon,F}$ forms a system of basic neighborhoods for $\phi$ for the weak-star topology (in the sense that, each $V_\phi(\varepsilon,F)$ is a neighborhood of $\phi$ and each neighborhood of $\phi$ contains a neighborhood of the form $V_\phi(\varepsilon,F)$).

With that in hand, assume that $\phi_n\to\phi$ in the sense that $\phi_n(x)\to\phi(x)$ for all $x\in X$. If we fix $\varepsilon>0$ and $F\subset X$ a finite set, then we can find $n_0$ so that for all $n\geq n_0$ we have that $|\phi_n(x)-\phi(x)|<\varepsilon$ for all $x\in F$. This shows that $\phi_n\in V_\phi(\varepsilon,F)$ for all $n\geq n_0$ and since $\{V_\varphi(\varepsilon,F)\}$ is a basic neighborhood system for $\phi$ for the weak-star topology, this shows that $\phi_n\to\phi$ in the weak-star topology.

Conversely, assume that $\phi_n\to\phi$ in the weak-star topology. Then, if $\varepsilon>0$ and $F\subset X$ is finite, we can find $n_0$ so that $|\phi_n(x)-\phi(x)|<\varepsilon$ for all $x\in F$. Fix $x_0\in X$ and let $\varepsilon>0$. Take $F=\{x_0\}$ and find such $n_0$, so you get $|\phi_n(x_0)-\phi(x_0)|<\varepsilon$ for all $n\geq n_0$, i.e. $\phi_n(x_0)\to\phi(x_0)$.