Dual space of locally convex space separates points

Question

Why does the following theorem fail if $X$ is not locally convex?

Let $X$ be a locally convex topological vector space. Then $X^*$ separates points.

I understand the proof and where it uses local convexity of $X.$ However, I fail to understand this intuitively. I would like some intuition on why this happens.

Edit: $L^p[0,1],0<p<1$ is not locally convex and it has no non-zero linear functional. So I do have a counter-example. I'm just looking for intuition.

Answer 1

Your counter-example provides an intuition. " The existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals." If the space is not locally convex, there might not be any(or sufficient) linear functionals to ensure separation.

Answer 2

Well, I'm not quite going to say that if $X^*$ separates points then $X$ must be locally convex, but it does follow that there are a lot of convex open sets in $X$.

For example, if $X^*$ separates points then the convex open sets separate points: Given $x\ne0$, choose $\Lambda\in X^*$ with $\Lambda x=1$. Let $V=\{y:|\Lambda y|<1/2\}$ and $W=\{y:|1-\Lambda y|<1/2\}$; then $V$ and $W$ are disjoint convex open sets, $0\in V$ and $x\in W$.