Dense subspaces of $L^p$ 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 $p \ne \infty$? 
 A: 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.
