Property of Dual Norm for a General Banach Space

Question

Let $E$ be a Banach space. Let $f \in E^*$. The dual norm is defined:

$$||f|| = \sup_{x \in E,\ ||x|| \leq 1} |\langle f , x\rangle |$$

Claim: We can find some $x \in E$ such that $||x|| = 1$, and $\langle f , x\rangle \geq \frac{1}{2} ||f||$.

Note that $E$ is a general vector space over some field $F$, not necessarily the reals. I have not come across this property before and an unsure how to prove it for a general field. Any hints appreciated.

Answer 1

If $\|f\|=0$ then $f=0$ and we can take any unit vector $x$. Otherwise $\frac 1 2 \|f\| <\|f\|$ so (by definition of $\|f\|$) there exists $y$ such that $\|y\| \leq 1$ and $\langle f,y \rangle \geq \frac 1 2 \|f\|$. Let $x=\frac 1 {\|y\|} y$. Then $\langle f,x \rangle = \frac 1 {\|y\|} \langle f,y \rangle\geq \frac 1 2 \|f\|$.