= Diagonal boundedness of operator families over real and complex Hilbert spaces
For operators on a complex <Hilbert space>, <pointwise> boundedness of $\langle Tx,x\rangle$ over a family implies uniform boundedness of their <operator norms>. The complex <polarization identity> recovers each $\langle Tx,y\rangle$ from four diagonal values. Apply the <Uniform boundedness principle> first to the functionals in $y$ for fixed $x$, then to the operators in $x$. On a real <Hilbert space> the claim fails: multiples of a nonzero skew-symmetric operator have zero diagonal forms but unbounded norms. For example, rotations through a quarter turn on $\mathbb R^2$ multiplied by integers suffice.
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