Consider the fractional Sobolev space $H^{1/2}\_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a\_0+\sum\_{k=1}^{\infty}(a\_k \cos kt+ b\_k\sin kt)$$ satisfy $$\sum\_{k=1}^{\infty}k(a\_k^2+b\_k^2)<\infty.$$
If we also consider $v\in H^{1/2}\_{2\pi}$ with the Fourier expansion $$v=\tilde{a}\_0+\sum\_{k=1}^{\infty}(\tilde{a}\_k\cos kt+\tilde{b}\_k\sin kt)$$ and define the inner product $(\cdot, \cdot):H^{1/2}\_{2\pi}\times H^{1/2}\_{2\pi}\to \mathbb{R}$ by $$(u, v)\_{H^{1/2}\_{2\pi}}=2\pi a\_0\tilde{a}\_0+\sum\_{k=1}^{\infty}k(a\_k\tilde{a}\_k+b\_k\tilde{b}\_k),$$ then this inner product defines a Hilbert space structure on $H^{1/2}\_{2\pi}$.


The fractional Sobolev space $H^{1/2}\_{2\pi}$ is frequently considered in the literature when looking for periodic solutions of Hamiltonian systems (to be precise, the space that one considers in this context is $E:=\left(H^{1/2}\_{2\pi}\right)^{2n}$, but there is nothing special about this space from a functional analytic perspective, so this is why I am considering the space $H^{1/2}\_{2\pi}$ in this question). I first encountered this space in the well known book *Minimax Methods in Critical Point Theory with Applications to Differential Equations* by Paul Rabinowitz when I started learning about the Mountain Pass Theorem and its generalisations. One particularly important result about this space that is frequently used in this context is the following:



> 
> For all $s\in [1, \infty)$, $H^1\_{2\pi}$ is compactly embedded in $L^s(0, 2\pi)$. In particular, there is an $\alpha\_s>0$ such that $$\left\lVert u \right\rVert\_{H^{1/2}}\le \alpha\_s \left\lVert u \right\rVert\_{L^s}.$$
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Rabinowitz does not prove this result in his book. He gives two references to which I unfortunately do not have access to:



> 
> A. Friedman, *Partial differential equations*, Holt, Rinehart, and Winston,
> Inc., New York, 1969.  
> 
> P. Rabinowitz, *A variational method for finding periodic Solutions of differential
> equation*, Nonlinear Evolution Equations (M. G. Crandall, ed.), Academic Press,
> New York, 1978 , pp. 225-251 .
> 
> 
> 


Does anyone know a modern reference for this result? The proof seems nontrivial, at least I do not know how to approach it. I could not locate a proof of this result in the literature even if it is widely used in the context of Hamiltonian systems that I mentioned before (I am not sure if the fact that the embedding is compact is ever used, but the fact that the embedding is continuous is ubiquitous).


P.S. I am sorry if this question is too easy for MO, but it did not seem fit for MS because to the best of my knowledge the space $H^{1/2}\_{2\pi}$ is pretty niche (at least I have not encountered it outside of this research area, but I am only a third year undergraduate student, so my assumption may be wrong due to my limited knowledge). Also please let me know if I should include any further details.



The compact embedding of $H^{1/2}_{2\pi}$ in $L^s(0, 2\pi)$
An excellent reference for this, with exercises, full details etc, is


*Demengel, Françoise; Demengel, Gilbert*, [**Functional spaces for the theory of elliptic partial differential equations. Transl. from the French by Reinie Erné**](https://doi.org/10.1007/978-1-4471-2807-6), Universitext. Berlin: Springer (ISBN 978-1-4471-2806-9/pbk; 978-1-4471-2807-6/ebook). xviii, 465 p. (2012). [ZBL1239.46001](https://zbmath.org/?q=an:1239.46001).


