- #1
logarithmic
- 103
- 0
Let C_b^\infty(\mathbb{R}^n) be the space of infinitely differentiable functions f, such that f and all its partial derivatives are bounded.
Is C_b^\infty(\mathbb{R}^n) dense in L^2(\mathbb{R}^n)? I think the answer is yes, because C_b^\infty(\mathbb{R}^n) contains C_0^\infty(\mathbb{R}^n), the space of all infinitely differentiable functions with compact support, as a subset. And it's well known that C_0^\infty(\mathbb{R}^n) is dense in L^p(\mathbb{R}^n).
However, there appear to be functions in C_b^\infty(\mathbb{R}^n) but are not in L^2(\mathbb{R}^n), for example the function f(x)=1. So this means that instead, we have C_b^\infty(\mathbb{R}^n)\cap L^2(\mathbb{R}^n) dense in L^2(\mathbb{R}^n)?
Is C_b^\infty(\mathbb{R}^n) dense in L^2(\mathbb{R}^n)? I think the answer is yes, because C_b^\infty(\mathbb{R}^n) contains C_0^\infty(\mathbb{R}^n), the space of all infinitely differentiable functions with compact support, as a subset. And it's well known that C_0^\infty(\mathbb{R}^n) is dense in L^p(\mathbb{R}^n).
However, there appear to be functions in C_b^\infty(\mathbb{R}^n) but are not in L^2(\mathbb{R}^n), for example the function f(x)=1. So this means that instead, we have C_b^\infty(\mathbb{R}^n)\cap L^2(\mathbb{R}^n) dense in L^2(\mathbb{R}^n)?