Let $T:H \to H$ be a bounded linear transformation between Hilbert spaces. Let $T^*$ be the adjoint of T. Show that $\|T\|=\|T^*\|$.
I know that $\langle Tx,y\rangle=\langle x,T^*y\rangle$, but I don't know how to use it to prove $\|T\|=\|T^*\|$.
2026-03-30 00:18:07.1774829887
Show that $\|T\|=\|T^*\|$
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Hint: Using the CBS inequality, you can obtain $$\|T\|=\sup\{|\langle Tx,y\rangle|:x,y\in H, \|x\|\leq1,\|y\|\leq1\}$$ from which the result follows rather quickly.