$\newcommand{\H}{\mathcal{H}}$Let $\H_1,\H_2$ be separable Hilbert spaces and $\mathcal{L}(\H_1,\H_2)$ the Banach space of bounded linear operators $\H_1\to\H_2$. Suppose $f:\mathbb{R}\to\mathcal{L}(\H_1,\H_2)$ in the weak operator topology, i.e., $\forall\,u\in\H_1$, $v\in\H_2$, we have that $x\mapsto\langle f(x)u,v\rangle_{\H_2}$ measurable in the usual sense. What can we say about $f$? Is $f$ strongly measurable in the sense that for any open set $U\subset\mathcal{L}(\H_1,\H_2)$ (in the norm topology), $f^{-1}(U)$ is measurable in $\mathbb{R}^n$?

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    $\begingroup$ Diestel and Uhl's book has examples to show that this is not true. (You can take $H_2$ to be one dimensional to make things simple). $\endgroup$ – Kavi Rama Murthy May 11 at 12:13

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