preserving problem

Question

Define: $$\begin{align*} \varphi &:L^2[0,1] \to L^2[0,1], \\ &(\varphi f)(x)=‎\int_0^x f(t)dt. \end{align*}‎$$

Is it true that, if ‎‎$‎‎B$ is a dense ‎subset ‎of ‎‎$‎‎L^2[0,1]$, then ‎$‎‎\varphi(B)$ ‎is dense in $‎‎L^2[0,1]$?

Answer 1

The range of the Volterra operator consists of all absolutely continuous functions vanishing at $0$. These are $L^2$-dense.

This also implies the (seemingly) stronger statement you gave.