By definition, $|T|=\sup|(Tf,g)|, |f|\le1,|g|\le1$
$$||T||\ge(Tf,f)$$
But I can not find an example such that $||T||>(Tf,f)$ for any $|f|<1$.
Any suggestion? Thanks in advance~
2026-04-04 04:34:36.1775277276
linear transformation between Hilbert space
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So, use the matrix suggested by Jose27 for the operator $T$, take a 2*1 vector (x,y) whose norm is bounded by 1, then ($T$(x,y),(x,y)) is ((0,x),(x,y))=xy has maximum 1/2. While the norm of $T$ is 1.
I am not sure what is the intuition of this, why the equality defines the norm does not hold when g=f?