It is true that $(B(H), SOT)$ is semireflexive, in which $H$ is a Hilbert space, and $B(H)$ is the set of all bounded linear operators from $H$ to $H$ with strong operator topology. As a starting point, I want to know whether $(B(H), SOT)$ is reflexive.
By the definition of reflexivity of a locally convex space, we attribute the above question to the following questions:
- Is $(B(H)^*, \sigma(B(H)^*,B(H)))$ semireflexive ? Thereinto, $B(H)^*=(B(H),SOT)^*$, and $\sigma(B(H)^*,B(H))$ is the weak star topology on $B(H)^*$.
- Is the strong operator topology SOT on $B(H)$ the strong topology $\beta(B(H),B(H)^*)$ on $B(H)$ ? That is, is this $SOT = \beta(B(H),B(H)^*)$ valid ?