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Let $H$ be a separable, infinite-dimensional Hilbert space and let $\mathbb{B}(H)$ be the algebra of bounded operators on $H$. The norm topolology on $\mathbb{B}(H)$ is stricly finer, hence the identity map from the strong to the norm topology is not continuous.

However, it is Borel measurable [This claim is false; see the comments. This relates to the fact that $\mathbb{B}(H)$ is not separable with respect to the norm topology]. To see this, let $v_1, v_2, \dots$ be a countable dense subset of the unit sphere of $H$. Then $$\{A \in \mathbb{B}(H) \mid \|A\|<1\} = \bigcap_{n \in \mathbb{N}} \{A \in \mathbb{B}(H) \mid \|Av_n\| < 1\}, $$ hence the norm unit ball is strongly measurable. The same is true for scalings and translates of the unit ball. Since these sets generate the norm topology, norm-open sets are strongly measurable. In total, we get that the Borel $\sigma$-algebra for the norm topology coincides with the $\sigma$-algebra for the strong topology.

Q: Is a similar statement true for the $\sigma$-algebra of sets with the Baire property?

Here a subset $S$ of a topological space has the Baire property if there exists an open set $U \subseteq X$ such that $S \Delta U$ is meagre; these sets form a $\sigma$-algebra.

So I wonder: Do meagre sets in the strong, respectively, norm topology coincide?

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    $\begingroup$ Since a norm open subset of $B(H)$ is not necessarily a countable union of open balls, norm open subsets need not be Borel in SOT. $\endgroup$
    – Narutaka OZAWA
    Commented May 26, 2021 at 13:25
  • $\begingroup$ Oh right, because $\mathbb{B}(H)$ is not separable in the norm topology... $\endgroup$
    – Matthias Ludewig
    Commented May 26, 2021 at 14:56
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    $\begingroup$ It is well exemplified by the following. Let $P_t$ denote the orthogonal projection from $L^2[0,1]$ onto $L^2[0,t]$. Then, $[0,1]\ni t\mapsto P_t\in B(L^2[0,1])$ is a homeomorphism w.r.t. SOT, but $\{P_t\}$ is discrete in the norm topology and hence any subset is relatively open. $\endgroup$
    – Narutaka OZAWA
    Commented May 27, 2021 at 0:25

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