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Is [itex]C_b^\infty(\mathbb{R}^n)[/itex] dense in [itex]L^2(\mathbb{R}^n)[/itex]? I think the answer is yes, because [itex]C_b^\infty(\mathbb{R}^n)[/itex] contains [itex]C_0^\infty(\mathbb{R}^n)[/itex], the space of all infinitely differentiable functions with compact support, as a subset. And it's well known that [itex]C_0^\infty(\mathbb{R}^n)[/itex] is dense in [itex]L^p(\mathbb{R}^n)[/itex].

However, there appear to be functions in [itex]C_b^\infty(\mathbb{R}^n)[/itex] but are not in [itex]L^2(\mathbb{R}^n)[/itex], for example the function f(x)=1. So this means that instead, we have [itex]C_b^\infty(\mathbb{R}^n)\cap L^2(\mathbb{R}^n)[/itex] dense in [itex]L^2(\mathbb{R}^n)[/itex]?