Let $\mathbb{H}$ be a (complex) Hilbert space. Define $\mathbb{P}$ as a projective space over $\mathbb{H}$, i.e. $\mathbb{P}=\left(\mathbb{H}\setminus\{0\}\right)/\sim$, where $f\sim g$ iff there exists $\lambda \in \mathbb{C}^{\times}$ s.th. $f=\lambda g$.
Define $U(\mathbb{H})$ the space of unitary transformations, i.e.
$U(\mathbb{H}):=\{U:\mathbb{H}\rightarrow \mathbb{H}, \ \mathbb{C}\mathrm{-linear, bijective}, \ \langle Uf,Ug\rangle=\langle f,g\rangle \ \forall f,g\in\mathbb{H}\}$
Similary we define $U(\mathbb{P})=\hat\gamma(U(\mathbb{H}))$, where $\hat\gamma$ is induced by $\gamma$, i.e. $\hat\gamma(U)(\varphi):=\gamma(U(f))$, for $U\in U(\mathbb{H}), \ \varphi=\gamma(f)\in\mathbb{P}$.
By strong topology I mean topology generated by subbasis
$\{U\in U(\mathbb{H}) \ | \ \|U_0(f)-U(f)\|<r \}$, for $U_0\in U(\mathbb{H}),\ r>0, \ f\in \mathbb{H}$.
If $\mathbb{H}$ is separable one can show that $U(\mathbb{P})$ and $U(\mathbb{H})$ are both metrizable (connected) topological groups.
My question : I'm looking for an examples for such nonseparable Hilbert spaces such that:
a) $U(\mathbb{H})$ and $U(\mathbb{P})$ are non-metrizable topological groups
b) $U(\mathbb{H})$ and $U(\mathbb{P})$ are non-metrizable and at least one of them is not a topological group
c) one of $U(\mathbb{H}), \ U(\mathbb{H})$ is a metrizable topological group but the other one is not.