Consider second quantization in physics, where $\mathscr{H}$ is the single particle Hilbert space and $\mathscr{F}$ is the corresponding Fock space. For simplicity, let it be the fermionic Fock space so that the creation, annihilation operators are bounded. Then is the algebra generated by creation/annihilation operators $c^*(f),c(f),f\in \mathscr{H}$ "dense" in the space of bounded linear operators $\mathscr{L} (\mathscr{F})$, where I'm being loose with "dense", in the sense that the underlying topology may be the norm-topology, strong-operator-topology, or weak-operator topology? Moreover, can any self-adjoint (possibly unbounded) operator be "approximated" by the ring under some topology or convergence?
EDIT: Assume that $\mathscr{H}$ is complex separable.