I am interested in studying the space $H^1([0,1])$. Using Fourier series, I am the. naturally led to studying the corresponding periodic Sobolev space. All references I can find on the internet deal with the case of a periodic Sobolev spaces $\mathbf{T}$. That is for $2 \pi$ periodic functions and not for arbitrary period periodic functions.
Can someone provide me with a reference where I can appropriate definitions, formulas etc. for this case?
There should be no difference between theorems and definitions for period $2\pi$ or just $2L$ for $L>0$: A function $f:\mathbb{R}\longrightarrow\mathbb{R}$ is $2\pi$-periodic, if and only if $f_L := f\left(\frac{2\pi}{2L}x\right) = f\left(\frac{\pi}{L}x\right)$ is $2L$-periodic.
So by using the substitution $x\longmapsto \frac{\pi}{L}x$, any statement about $2\pi$-periodic functions can be transformed into a statement about $2L$-periodic functions. Of course, if there are derivatives of $f$ involved, one has to also use the chain rule.