Dense subspaces of $L^p$

Question

Let $B$ a separable Banach space and $\nu$ a Borel probability measure on $B$. Let us consider the space $C^1_b(B)$ of the continuously differentiable functions bounded and with bounded derivative. Is it true that $C^1_b(B)$ is dense in $L^p(B,\mu)$ for every $p \ne \infty$?

Answer 1

This question is more for math overflow, I think. I only have a very partial answer. Theorem 4.13 in the book Geometric Nonlinear Functional Analysis by Benyamini and Lindenstrauss says that if $B$ is a Banach space and its dual $B^{\prime}$ is separable, then it admits an equivalent norm which is Frechet differentiable except at the origin. Using that you can construct $C^{1}$ bump functions and $C^{1}$ partitions partitions of unity.

Instead if $B$ is separable, then $E$ has an equivalent norm which is only Gateaux differentiable, so in this case there might not be bumb functions I guess.

Then in Corollary 4.14 it shows that if $B$ is a Banach space and its dual $B^{\prime}$ is separable, then every continuous function $f:U\rightarrow \mathbb{R}$, where $U$ is an open subset of $B$, can be uniformly approximated by a $C^{1}$ function. I guess one could use this first approximate $L^p$ functions with continuous functions and then continuous with $C^1$. Don't know about $C^\infty$. This corollary is due to Bonic and Frampton. [1] It is an interesting paper. It discusses the existence of bump functions. As Martins Bruveris wrote, if there are no bump functions, I very much doubt a characteristic function can be approximated by a smooth function.