Let \(\mathcal{H}\) be a real Hilbert space.

(i) Show that

\[\|h\|=\sup _{g eq 0} \frac{|\langle g, hangle|}{\|g\|}=\sup _{\|g\| \leqslant 1}|\langle g, hangle|=\sup _{\|g\|=1}|\langle g, hangle|\]

(ii) Can we replace in (i) \(|\langle\cdot, \cdotangle|\) by \(\langle\cdot, \cdotangle\) ?

(iii) Is it enough to take \(g\) in (i) from a dense subset rather than from \(\mathcal{H}\) (resp. \(\overline{B_{1}(0)}\) or \(\{k \in \mathcal{H}:\|k\|=1\}\) )?