$H$ is an Hilbert space in which $\{h_n\}_1^\infty$ is weakly converges, I have a problem understanding a step in the proof for the proposition which claims that $\sup_{n\in \mathbb{N}}\|h_n\| < \infty$ i.e $\{h_n\}_1^\infty$ is bounded.
The proof defines $F_n:= \{g\in H : \forall k\in \mathbb{N} : \frac{|\langle g,h_k|\rangle}{\|g\|} \le n \}$ and shows that $F_n$ is a closed set in $H$ and $H = \bigcup_{n\in \mathbb{N}} F_n$. So , Baire's category theorem promises that there is $n_0$ such that $\mathrm{int}(\overline {F_{n_0}}) = \mathrm{int}(F_{n_0})\neq \emptyset$ (since $F_n$ are closed) thus $\exists g_0\in F_{n_0} , \exists r>0 : \overline{B(g_0,r)}\subseteq F_{n_0}$. Since $\overline{B(g_0,r)} = \{g_0 +f : \|f\|\le r\}$ we get the first bound:
$\forall f\in \overline{B(0,r)}:|\langle f,h_k \rangle| =|\langle f+g_0,h_k \rangle - \langle g_0,h_k \rangle| \le |\langle f+g_0,h_k \rangle |+| \langle g_0,h_k \rangle|$
$\le \|f+g_0\|n_0 + \|g_0\|n_0 \le _0(2\|g_0\| + \|f\|) \le n_0(2\|g_0\| + r).$
Now for the problematic part, the proof I was introduced with claim that $\sup_{f\in \overline{B(0,r)}} |\langle f,h_k\rangle| =r\cdot \|h_k\|$ but I don't understand how does one show the equality , the inequality $\sup_{f\in \overline{B(0,r)}} |\langle f,h_k\rangle| \le r\cdot \|h_k\|$ can be showed by Cauchy Schwartz theorem, but I don't see how to show this equality.
Just i order to finish the proof: with that equality one get that $\forall k \in \mathbb{N} : \|h_k\| \le n_0 (\frac{2\|g_0\|}{r} + 1)$.
If $h_k=0$ this is trivial. Assume $h_k \neq 0$. Let $f=\frac {r h_k} {\|h_k\|}$. Then $\|f\| \leq r$. Hence $\sup \{\langle f, h_k \rangle: f \in \overline { B(0,r)}\} \geq \langle f, h_k \rangle $ Compute this last expression to finish the proof.