Continuous and dense embeddings and the density of sets in Hilbert space.

Question

Suppose $H$ is a Hilbert space of functions $f:\Omega\to \mathbb{R}^n$ with $\Omega\subset \mathbb{R}^n$ open, bounded and with Lipschitz boundary (take for example $H=H_0^1(\Omega)^n$) and suppose $B$ is a Banach space that is continuously and densely embedded in $H$. Let $g\geq 0$ and continuous, is the set $$C(B,g):=\{f\in B: |f(x)|_{\mathbb{R}^n}\leq g(x) \text{ a.e. } x\in \Omega\}$$ dense in $$C(H,g):=\{f\in H: |f(x)|_{\mathbb{R}^n}\leq g(x) \text{ a.e. } x\in \Omega\}?$$