Intuition behind : for all $x\in E$ there is $f_0\in E^*$ s.t. $\left=\|x_0\|^2$ and $\|f_0\|=\|x_0\|$.

Question

I'm seeing a theorem that say that for all $x\in E$ there is $f_0\in E^*$ s.t. $\left<f_0,x_0\right>=\|x_0\|^2$ and $\|f_0\|=\|x_0\|$. I recall that $E^*$ denote the topological dual of $E$. Is there a similar result for inner product spaces for example ? I don't really see the intuition behind this proposition.

Answer 1

In a Hilbert space $H$ you can simply take $f_0 = x_0$ (using the identification of $H^*$ with $H$ given by the Riesz representation theorem).

Your theorem says that you can find a functional acting in the same way also in a normed vector space.