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Let $H$ be a nondegenerate, not positive definite, Hermitian form on a complex vector space $V$ such that $$|H(x,y)|\le u(x)u(y)\tag{B}$$ for some map $u:V\to R_+$ with $u(\lambda x)=|\lambda|u(x)$ for $u\in V$, $\lambda \in C$.

Clearly, condition $(\textrm{B})$ is necessary for the existence of a Euclidean norm on $V$ (defined in the standard way from a positive definite Hermitian form on $V$) such that $$|H(x,y)|\le \|x\|\,\|y\|.$$ Condition $(\textrm{B})$ is sufficient (and indeed superfluous) in the finite-dimensional case. But what happens in infinite dimensions?

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  • $\begingroup$ When the omission of a hypothesis is central to a statement it is a good idea to emphasize it. It would have saved me a lot of time if the question said "Let $H$ be a not necessarily positive Hermitian form..." $\endgroup$
    – Ruy
    Commented Sep 27, 2020 at 18:42

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Since I posed the question I found that the book

  • J. Bognár, Indefinite inner product spaces, Springer 1974

has a counterexample in infinite dimensions; see his Example 5.6, p.90.

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