I was reading a book on harmonic analysis and I found the following definiton:
If $a\in\mathbb{R}$ we define the sobolev space $L_a^2(\mathbb{R})$ as:
$$ L_a^2(\mathbb{R}^n)=\lbrace g:L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)\mid (1+|\xi|^2)^{a/2}\hat{g}(\xi)\in L^2(\mathbb{R}^n)\rbrace$$
But, the definition I was used to see in the books is the following: $$L_k^2(\mathbb{R}^n)=\lbrace g:L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)\mid \partial^\alpha g\in L^2(\mathbb{R}^n)~\forall~|\alpha|\leq k\rbrace$$ where $k$ is a positive integer. My question is if the first definition is equivalent to the second one in the case when $a$ is a positive integer. It must be, since otherwise it wouldn't be called sovolev space, but I don't see how to prove it.
Any help would be thanked.