The Volterra operator $V:L^{2}[0,1]\rightarrow L^{2}[0,1]$ is defined by $(Vf)(x)=\int_0^xf(t)dt$.
I am wondering if it can be shown that $V$ is compact by definition - that is, either that $V$ maps bounded sets to precompact sets, or equivalently, that for any bounded sequence $(f_n)$ in the domain, $\{Vf_n\}$ has a convergent subsequence.
I have seen an elegant proof of the compactness of $V$ using the Arzela-Ascoli theorem. Also, I have come across proofs using the notion of Hilbert-Schmidt operators. However, both of these notions were foreign to me when I was assigned this problem, and I am curious if I can show $V$ is compact without using these ideas; unfortunately, I am not sure how to proceed directly.
Thank you.