consequence of Hahn-Banach theorem In Wikipedia, it says that Hahn-Banach Theorem shows there are "enough" continuous linear functionals. But, why is that so in a space that is not necessarily normed? How does the statement of Hahn-Banach show this?

Let $X$ be a normed space and $M$ a finite-dimensional subspace of $X$. Take any linear functional $f : M \to \mathbb{F}$. It will be continuous because $M$ is finite-dimensional.
Hahn-Banach lets you extend $f$ to a continuous linear functional defined on all of $X$, so you obtain a nontrivial element of $X^*$.
