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I want to show that for $E$ a normed space, $f\in E^*$ and $M=\{x\in E\,:\, f(x)=0\}$:

  1. Write $M^\perp$
  2. Show that $d(x,M)=\frac{|\langle f,x\rangle|}{||f||}$.

This is my attempt: For the second part: We have that $f\in E^*$ and for $x\in E$ and $m\in M$ $$\ |\langle f,x-m|\rangle \leq ||f|| ||x-m|| \Rightarrow \frac{|\langle f,x\rangle|}{||f||} \leq ||x-m||. $$ Therefore, $$ \frac{|\langle f,x\rangle|}{||f||} \leq \inf_{m\in M}||x-m|| =d(x,M). $$ The second inequalyty is that I can't prove, I think that any corollary of Hanh-Banach could help me to prove that $$d(x,M)\leq \frac{|\langle f,x\rangle|}{||f||} $$ Does anyone have any idea and could check my proof?

Update

I found the same question in this link Orthogonality Relations Exercise, Brezis' Book Functional Analysis