Property of bounded linear transformation between Hilbert spaces I've asked a question on related question in a previous thread, but I wanted to ask a follow up question.

If a bounded linear transformation $T: X \to Y$ where $X$ and $Y$ are Hilbert spaces has closed range, show that the set $B = \{f \in \text{Ran}(T): \|T^*f\|_X \leq 1 \} $ is bounded.

I was going to first use surjectivity of the closed range to prove $B$ is weakly bounded, and go from there, but I have no how to prove weak bounded and also how to  go from there. Thanks!

Say $R=T(X)$. Then $R$ is a closed subspace of $Y$, so $R$ is a Hilbert space. The Open Mapping Theorem applied to $T:X\to R$ shows that there exists $\delta>0$ such that for every $f\in R$ there exists $g\in X$ with $Tg=f$ and $\|g\|\le\delta\|f\|$.
Now suppose $f\in R$, $\|T^*f\|\le 1$ and choose $g$ as above. Then $$\|f\|^2=\langle f,f\rangle=\langle Tg,f\rangle =\langle g,T^*f\rangle \le\|g\|\le\delta\|f\|.$$
