Let $\mathbb{B}$ be an infinite-dimensional Banach space. I want to prove that every nonempty weak* open subset $\mathbb{O}$ of $\mathbb{B^*}$ is unbounded.
Without loss of generality, let's assume $0 \in \mathbb{O}.$ If this not true, we just do some translation. Now, by the definition of the weak∗ topology, $\mathbb{O}$ contains an open set $\mathbb{U}$ of the form $\{f:|f(x_i)|<\epsilon, 1≤i≤n\}$ for some $\epsilon >0$ and $x_i \in \mathbb{B}.$ This an exercise from Folland's book. I am trying to prove it using Hahn Banach Theorem. I know a couple of corollaries of the Hahn Banach Theorem, however, I cannot use any of the corollaries I know so far. Could you give me some hints so that I can apply the Hahn Banach theorem to solve this problem?
Thank you so much. Any help/suggestions would be highly appreciated.
Let $M$ be the span of $x_1,x_2,..,x_n$. Let $y$ be any vector not in $M$ and $N=span (M \cup \{y\})$. Define a linear functional $f$ on $N$ such that $f(x_i)=0$ for each $i$ and $f(y)=1$. [ You can do this by taking a basis for $M$ and attaching $y$ to it to get basis for $n$]. Recall that any linear map on a finite dimensional space is continuous. Now $f$ extends to a continuous linear functional on $\mathbb B$ such that $f(x_i)=0$ for each $i$ but $f(y)=1$. Now note that $nf$ is in your neighborhood for each positive integer $n$. Since $f(ny)=n \to \infty$ we are done.