I understand that this question would be trivial for experts, sorry for that, I just need to clarify things.


So let $S(\mathbb{R}^n)$ denote the Schwartz space on $\mathbb{R}^n$ and $W\_p$, $W\_q$ are the Sobolev spaces, or in the other words completions of $S(\mathbb{R}^n)$ with respect to $p$ and $q$ - Sobolev norms. 


Let us assume that $\phi: $ $W\_q\rightarrow W\_p$ is a map , continuously extending the identity map on $S$ assuming that $p < q$.


Question: why is $\phi$ an embedding? It is only clear that $\phi$ is embedding of $S$ ( as it is identity on it).


Similar question for Sobolev embedding theorem into $C^k$.


P.S. To clarify my question here is what I mean by the Sobolev space. For a real number $p\in\mathbb{R}$ the $p$-Sobolev norm on $S$ is given by 
${|f|}^2\_p=\int{(1+|\xi|)}^{2p}{|\hat{f}(\xi)|}^2d\xi$, where $\hat{f}(\xi)$ is a Fourier transform of $f(x)\in S$. So $W\_p$ is a completion of $S$ with respect to this norm.


P.P.S. To make things little more clear I am mostly interested in case of non-integer or negative $p$ when the Sobolev $p$-norm can not be defined through the weak derivatives.



Easy question on Sobolev spaces
I would like to expand a bit what Delio said. Your question is a bit confusing as it is, but we may assume that you mean that $p$ and $q$ are the parameters representing the derivatives. Then what you need is
$$ \|f\|\_p \leq C\|f\|\_q.$$


If you write out the definition of the Sobolev norm on $S$, then you see immediately that this holds. This implies that the identity map defined on $S$ extends uniquely and continuously to the whole $H^q$. This gives you the embedding. (Strictly speaking you have to verify that this extension is the identity map, but this follows easily from continuity considerations.)


