I am doing a course on functional analysis. One of the pillars in functional analysis is Hahn-Banach theorem. But in different books I find different versions of the theorem and it is not easy to check that they are equivalent. So, I want to understand why Hahn-Banach theorem is important. Also the proof uses Zorn's lemma which is equivalent to axiom of choice, so I cannot understand it by concrete examples. The statement given by our instructor for Hahn-Banach theorem is as follows:
Let $X$ be a normed linear space and $X_0$ be a subspace of $X$,let $f_0$ be a bounded linear functional on $X_0$,then there is a bounded linear functional $f$ on $X$ such that $f|_{X_0}=f_0$ and $||f||=||f_0||$ i.e. $\sup\limits_{||x||=1,x\in X}|f(x)|=\sup\limits_{||x||=1,x\in X_0}|f_0(x)|$.
I want to understand the usefulness of the fact that $||f||=||f_0||,$ i.e. its physical significance and why it is good to have such an extension. Can someone give me some idea?