Given is a normed (and not necessarily complete) vector space $(X, ||\cdot||)$, a linear map $L:X\to X$ and a subset $D\subset X$. What conditions must $D$ satisfy such that $L(D)$ is dense in $L(X)$?
This problem has the following background: Let $X=Lip([c,d])$ (the set of all Lipschitz-continuous functions $[c,d]\to\mathbb R$) with the norm $||f||=|f(a)|+Var(f)$ (where $Var$ denotes the classical variation of a function $[c,d]\to\mathbb R$) and let $D=C^1([c,d])$, then $L_f:Lip([c,d])\to Lip([c,d]),\ g\mapsto g\circ f$ is linear for each $f:[a,b]\to[c,d]$ of bounded variation. I want to know if for each such $f$ and each $\epsilon>0$ and each $g\in Lip([c,d])$ there is some $h\in C^1([c,d])$ such that $||g\circ f-h\circ f||<\epsilon$.
Any help is highly appreciated. Thanks in advance!