I saw that it was already asked, but the book where I'm studying is slightly different. Recall some definition, if $E$ it's $\mathbb{K}$-vector space and let $\mathcal{E}$ be a vector subspace of the algebraic dual of $E$, wich is the vector space of all linear forms on $E$. We say that $(E,\mathcal{E})$ is a dual couple if $\mathcal{E}$ separates points of $E$, that is, $u(x)=u(y)$ $\forall u \in \mathcal{E}$ implies $x=y$. We have the following result:
- "Suppose $(E,\mathcal{E})$ is a dual couple , $u_1,...,u_n , u \in \mathcal{E}$. Then $u=\sum_{k=1}^n \lambda_k u_k$ iff $u_1(x)=...=u_n(x)=0$ implies $u(x)=0$"
Exercise (a): Let $E$ be an infinite-dimensional locally convex space. Prove that the weak* topology $\sigma(E',E)$ on $E'$ is metrizable if and only if $E$ has a countable algebraic basis.
Hint (a): Let $U_n=\lbrace \max\lbrace p_{x_1},...,p_{x_N} \rbrace < 1/M \rbrace$. If $x \in E$, every ball $\lbrace p_x < \epsilon \rbrace$ contains some $U_n$; that is, $|u(x_j| < 1/n$ $\forall j$ implies $|u(x)| <1$. Hence $u(x_1)= \cdot \cdot \cdot u(x_N)=0$ implies $u(x)=0$ and $x \in \mathrm{span}(x_1,...,x_n)$.
Proof (Exercise (a)).
We assume that the weak* topology $\sigma(E',E)$ on $E'$ it's metrizable. Let $x_1,...,x_N \in E \setminus \lbrace 0 \rbrace$. Note that the weak* topology is defined by seminorm family $\mathcal{F}= \lbrace p_x(u)=|u(x)| : x \in E \rbrace$, where
\begin{align*}
\displaystyle U_n := \lbrace u \in E' : \max \lbrace p_{x_1}(u),...,p_{x_N}(u) \rbrace < 1/n \rbrace
\end{align*}
It is an open neighborhood in weak* topology.
By hypothesis $\forall x \in E$, $\exists n \in \mathbb{N}$ and exists open ball $B_{E'}:=\lbrace u \in E' : p_x(u):=|u(x)| < \epsilon \rbrace$ such that $U_n \subset B_{E'}$.
Now, if $u \in U_n$ we have $|u(x_j)| < 1/n$ $\forall j$ and then $|u(x)|< 1$ with $\epsilon=1$, and by (1) we have:
\begin{align*}
u(x_1)=...=u(x_N)=0 \Longrightarrow 0=u(x)=\sum_{j=1}^n \lambda_j u(x_j)
\end{align*}
by linearity $x \in \mathrm{span}(x_1,...,x_n)$.
Reciprocally, just reverse the above implications, using results in (1)?
Someone can help me? can you check if proof of exercise (a) is correct? Thank you.