Norm of linear continuous functions on a Banach space

Question

I have this lemma, is it correct like this or i must say $$\sup_{\langle g,y\rangle=0, \|y\|=1}|\langle f,y\rangle|=\min_{\lambda\in\mathbb{R}}\|f-\lambda g\|$$

Answer 1

If $X$ is a complex banach space, you will need to take the absolute value, since $\langle f,y \rangle$ is maybe a complex number. On the other, if $X$ is real, everything is okay! (One obvious reason is that we can replace $y$ by $-y$ if $\langle f,y \rangle$ is negative.)