I am reading one paper and I am stuck at one point about their conclusion.
So, $V$ and $W$ are Hilbert spaces. $T$ is a linear map between them whose adjoint exists. So, we have $\langle Tx,y\rangle=\langle x,T^*y\rangle$. So, for all $x\in V$ such that $\|x\| \leq 1$, and for all $y\in W$, we have $|\langle Tx,y\rangle| \leq \|T^*y\|$. (This is by Cauchy-Schwarz inequality). The problem lies in their next conclusion which I am confused about. It says "Consequently, it follows from the uniform boundedness principle that $\underset{\|x\|\leq 1}{\sup}\|Tx\| < \infty"$. I am confused about how they invoke the PUB here. Can someone please explain in detail.
For each $x$ with $\|x\| \leq 1$ define $T_x(y)=\langle y, Tx \rangle$ This is a continuous linear functional. For fixed $y$ we have $|T_x(y)| \leq \|T^{*}y\|$ so $\sup_x |T_x(y)| <\infty$. UBP now implies that $\sup_x \sup_{\{\|y\| \leq 1\}}| T_x(y)| <\infty$ which gives $\sup_x \|Tx||<\infty$. [I am writing $\sup_x$ for sup over $x$ with $\|x\|\leq 1$].