On page 59, in the first part of the geometric form of Hahn-Banach theorem, it only assumed that $X$ is a topological space and it says
there exists $\Lambda\in X^*$ and ...
Here $X^*$ is the continuous dual.
But we know when the topological vector space is not locally convex, the continuous dual space might be trivial, $L^p[0,1]$ where $0<p<1$ is such example. How is this possible?
In the second part of the theorem, we have the additional assumption that $X$ is locally convex.
The theorem never in any way asserts that a nontrivial element of $X^*$ must exist for arbitrary $X$. It only asserts that given disjoint, nonempty convex sets $A,B\subset X$ with $A$ open, then there exists a $\Lambda\in X^*$ satisfying the stated inequalities. It's entirely possible that for some $X$, no such sets $A$ and $B$ exist.