TITLE: Hahn Banach Theorem
QUESTION [7 upvotes]: It is stated often that the Hahn Banach Theorem makes the study of the dual space "interesting". What does this exactly mean though?
I.e what is exactly meant by "interesting"?
I am puzzled as to why it follows immediately from Hahn-Banach that the dual of a (non-zero) normed vector space is non-trivial.
How does it follow DIRECTLY from Hahn Banach that there are non-trivial functions?

REPLY [15 votes]: A consequence of Hahn Banach is that linear functionals separate points. This implies a certain richness of the space of linear functionals.
Separating points means that given two distinct points $x$ and $y$ there is a continuous linear functional $f$ such that $f(x)\neq f(y)$.
Added in light of your inquiry: to prove that there is such a functional, consider the one-dimensional subspace $\mathbb{C}(x-y)$ (complex multiples of $x-y$). You can easily show that on this subspace $f(\lambda (x-y))=\lambda||x-y||$ defines a continuous linear functional. You can then extend this to your whole space by Hahn-Banach and by linearity it will follow that $f(x)-f(y)=f(x-y)=||x-y||\neq 0$, so $f(x)\neq f(y)$, as desired.