Closed Hilbert half-space

Question

Suppose $H$ is the seperable infinite-dimensional real Hilbert space and $f$ is a continuous linear functional on it. Is the closed half-space $H_{f \ge 0} = \{ x \in H | f(x) \ge 0 \}$ homeomorphic to $H$? They are both contractible even after taking the complement of any compact subset, so the ususal proof for finite-dimensional $\mathbb{R}^n \ncong \mathbb{R}^n_{\ge 0}$ does not extend. Moreover, the famous theorem that 'Hilbert manifolds of the same homotopy type are homeomorphic' is not valid either, since $H_{f \ge 0}$ being a Hilbert manifold is precisely equivalent to $H \cong H_{f \ge 0}$. Any reference is welcome.