By Dirichlet space we mean $\{F\in C_{0}[0,1]:\text{ there exists }f\in L^{2}[0,1]\text{ with }F(t)=\int_0^t f(x) \, dx$, $\forall t\in [0,1]\}$.
The more the better. Any famous examples?
Using the basis of $L^2$[0,1] : $F(t)=\sum_{n=-\infty}^\infty a_n \frac{e^{2\pi i n t}-1}{2\pi i n}$ for $a_n\in [0,1]$ (given that the sum converges and so I can swap sum and integral). Or using the legendre polynomials $F(t)=\sum_{n=0}^\infty a_n \frac{t^{n+1}}{n+1}$
On second thought, this question is a bit silly since all continuous functions on $[0,1]$ are a subset of $L^2$[0,1]. But I will leave it here for other weary travelers.
Thanks