Is $L^\infty(0,T;V)$ a reflexive space? Question about weak convergence

Question

Let $V$ an Hilbert space and $T>0$. Is $L^\infty(0,T;V):=\{v:[0,T]\to V: \text{ess}\,\text{sup}_{t\in [0,T]}||u(t)||<\infty$ a reflexive space?

I think that since the $L^\infty$ isn't reflexive, then $L^\infty(0,T;V)$ couldn't be reflexive too.

I'd like that any bounded sequence in $L^\infty(0,T;V)$ converges weakly, but since this space is not reflexive, i can' get this.

Am I right? Thanks for your help

Answer 1

Any closed subspace of reflexive space is reflexive. $L_\infty[0,T]$ is not reflexive but embedded in $L_\infty([0,T],V)$ as closed subspace via the map $i(f)(t)=f(t)x$, where $x$ any norm one vector in $V$.