Nomenclature in this area of mathematics is so confusing, I cannot even determine what I am to prove here. (Weak convergence can mean about three things, and so can boundedness.) Is it
$$f_n \xrightarrow{WOT}f \Rightarrow (\exists M ∈ ℝ_{≥ 0})(\forall n ∈ ℕ)(\lVert f_n \rVert ≤ M)? $$
In all three books I use (Rynne & Youngson, Conway, and Murphy), 'norm-bounded' is nowhere defined. I guess for analysts this may be obvious from the context, but for me it isn't.
Potential answer. (I've since been told that my definition of norm-boundedness was the right one.)
We have \begin{align*} \lim_{n → \infty} \lVert f_n \rVert &= \lim_{n → \infty} \sup\{\lVert f_n(h) \rVert\mid \lVert h \rVert≤ 1\} = \lim_{n → \infty} \sup\{\sqrt{\langle f_n(h), f_n(h) \rangle} \mid \lVert h\rVert ≤ 1 \} \\ &= \sup\{\sqrt{\langle f(h), f(h) \rangle} \mid \lVert h\rVert ≤ 1 \} \\ &= \sup\{\lVert f(h) \rVert \mid \lVert h\rVert ≤ 1 \} = \lVert f \rVert. \end{align*}
Now I would guess this implies boundedness of the sequence (if there are $n ∈ ℕ$ with $\lVert f_n \rVert > \lVert f \rVert$ it can only be finitely many, say, some finite $S \subset ℕ$, say, and we can take $M := \max(\{\lVert f \rVert\}∪\{\lVert f_n\rVert\ \mid n ∈ S\})+1$, for instance.
Problem is that I rarely seem to know which steps are allowed, why they are allowed, and which verifications actually need to be done.. The argument I presented above seems convincing, but please feel free to correct/add anything I may have missed.