I am new to Sobolev space and distribution theory. So here is what I know. Distributions are linear functionals on $C_0^\infty$. Let's look at the simplest Sobolev space. $H^1(\Omega)$ is equal to the closure (with respect to $L_2$ norm) of the following set $$\{f\in C(\Omega): \int (1+|\xi|^2)|\hat{f}(\xi)|^2d\xi<\infty\}$$ where $\hat{f}$ is the Fourier transform of $f$.
Now there is this claim that $H^1(\Omega)$ is the subspace of $L_2$ such that the distributional derivative is still in $L_2$. I am a little confused about this statement. Since distributional derivative is a distribution, what does it mean for a distribution to be in $L_2$?
A distribution $\mu$ is said to be in $L_2$ if there exists a function $f \in L_2$ so that for all test functions $\phi$ we have $\mu(\phi) = \int_{\mathbb R} f(x) \phi(x) \, dx$.