A Problem when studying nonlinear approximation

Question

As an undergraduate, I've chosen Neural network as final year project, so nonlinear approximation theory is a main requirement to learn it. However, when I start learning approximation, the textbook/paper has mentioned several times about Banach space, Sobolev space, $L_p$ norm, error estimation bounded below $Mn^{-r}$. There are always two formulations for error, one feels standard for me: $$E_n(f)\leq Mn^{-r},$$ another is in the binary sense: $$\text{ quasi-norm for $\mathscr A_q^r$}:\quad |f|_{\mathscr A_q^r(X)}:= \lVert (2^{kr}E_{2^k}(f))\rVert_{l_q}.$$ I'm not really familiar with these notations, which I feel like I've missed something in between, probably more elementary than that. I feel like that also because the text provides many different formulation, which I couldn't get their motivations, like why they concern about smoothness, why using binary form to express error, why using $L_p$ norm instead of $L_2$, and so on. I knew Fourier series and relevant error approximation theory very well, have the knowledge of Mathematical analysis, topology, advanced linear algebra, abstract algebra, and other fields I think not relevant to approximation theory. Can you suggest me what are the more elementary books to prepare for this approximation theory? Similar books with more friendly notations will also be great.