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

Clearly, condition (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 (B) is sufficient (and indeed superfluous) in the finite-dimensional case. But what happens in infinite dimensions?

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?

Let $H$ be a nondegenerate Hermitian form on a complex vector space $V$ such that $$|H(x,y)|\le u(x)u(y)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (B)$$ for some map $u:V\to R_+$ with $u(\lambda x)=|\lambda|u(x)$ for $u\in V$, $\lambda \in C$.

Clearly, condition (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 (B) is sufficient (and indeed superfluous) in the finite-dimensional case. But what happens in infinite dimensions?

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

Clearly, condition (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 (B) is sufficient (and indeed superfluous) in the finite-dimensional case. But what happens in infinite dimensions?

Let $H$ be a nondegenerate Hermitian form on a complex vector space $V$ such that $$|H(x,y)|\le u(x)u(y)$$ for some map $u:V\to R_+$ with $u(\lambda x)=|\lambda|u(x)$ for $u\in V$, $\lambda \in C$.

Clearly, this 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 $$|H(x,y)|\le \|x\|\,\|y\|.$$ The condition is sufficient (and indeed superfluous) in the finite-dimensional case. But what happens in infinite dimensions?

Let $H$ be a nondegenerate Hermitian form on a complex vector space $V$ such that $$|H(x,y)|\le u(x)u(y)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (B)$$ for some map $u:V\to R_+$ with $u(\lambda x)=|\lambda|u(x)$ for $u\in V$, $\lambda \in C$.

Clearly, condition (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 (B) is sufficient (and indeed superfluous) in the finite-dimensional case. But what happens in infinite dimensions?