I am reading Rudin's Functional Analysis and got quite confused by his proof for theorem 7.25, which he calls Sobolev's Lemma.
In proving the theorem, he defines the function $F$, and calculates its distributional derivative $D_i^k F$ (eq.3). Then he says this implies $D_i^k F$, originally defined as a distribution, is actually a $\mathcal{L}^2$.
This is what confuses me because $D_i^k$ is only the distributional derivative, how can one identify this with the function derivatives? Maybe we can redefine $F$ so that the distributional derivative agrees with the classical derivative but I still feel unconfortabla unless someone points out how this can be done without contradicting our original definition.
Thanks!