There is no inner product that induces the infinity norm in the space of continuous complex functions in [a,b]

Question

There is no inner product that induces the infinity norm $\|f\|_{\infty}:=\sup\{｜f(x)｜: x \in [a,b] \}$ in the space of continuous COMPLEX functions of domain $[a,b]$, where $[a,b]$ is a closed interval of the real numbers.

I couldn't prove this and I couldn't find counterexamples for the parallelogram rule. Can you provide a proof of this statement?

Answer 1

You can take$$f_1(x)=\begin{cases}\frac{\frac{a+b}2-x}{b-a}&\text{ if }x\leqslant \frac{a+b}2\\0&\text{ if }x>\frac{a+b}2\end{cases}\quad\text{and}\quad f_2(x)=\begin{cases}0&\text{ if }x\leqslant \frac{a+b}2\\\frac{x-\frac{a+b}2}{b-a}&\text{ if }x>\frac{a+b}2.\end{cases}$$Then all of the functions $f_1$, $f_2$, $f_1+f_2$, and $f_1-f_2$ have norm $\frac12$. Therefore, the parallelogram law doesn't hold.