Concepts in mathematics which are referred to as 'generalizations' I am curious to know some theorems usually taught in advanced math courses which are considered 'generalizations' of theorems you learn in early university or late high school (or even late university).
For example, I know that Stokes's theorem is a generalization of the divergence theorem, the fundamental theorem of calculus and Green's theorem, among I'm sure many other notions.
I've read that pure mathematics is concerned mostly with the concept of 'generalization' and I am wondering which theorems/ideas/concepts, like Stokes's theorem, are currently celebrated 'generalizations' by  mathematicians.

Cauchy-Schwarz inequality tells us that ($\forall a_i,b_i\in\mathbb R$, $i\in\{1,2,\ldots,n\}$):
$(a_1^2+a_2^2+\cdots+a_n^2)(b_1^2+b_2^2+\cdots +b_n^2)\ge (a_1b_1+a_2b_2+\cdots+a_nb_n)^2$
Whereas Hölder's inequality lets us know that ($\forall a_{ij}\in\mathbb R^{+}$, $i\in\{1,2,\ldots,m\},j\in\{1,2,\ldots,n\}$):
$(a_{11}^m+a_{12}^m+\cdots+a_{1n}^m)(a_{21}^m+a_{22}^m+\cdots+a_{2n}^m)\cdots(a_{m1}^m+a_{m2}^m+\cdots+a_{mn}^m)$
$\ge \left(a_{11}a_{21}\cdots a_{m1}+a_{12}a_{22}\cdots a_{m2}+\cdots+a_{1n}a_{2n}\cdots a_{mn}\right)^m$
The latter inequality is just a special case Hölder's inequality. The actual one states that
$\forall a_{ij}\in\mathbb R^{+}$, $i\in\{1,2,\ldots,m\},j\in\{1,2,\ldots,n\}$ and
$\forall p_i\in\mathbb R^+, i\in\{1,2,\ldots,m\}, p_1+\cdots+p_m=1$  
$$(a_{11}+\cdots+a_{1n})^{p_1}\cdots(a_{m1}+\cdots+a_{mn})^{p_m}\ge a_{11}^{p_1}a_{21}^{p_2}\cdots a_{m1}^{p_m}+\cdots+a_{1n}^{p_1}a_{2n}^{p_2}\cdots a_{mn}^{p_m}$$
This one requires positive reals, so I put this requirement into the special case too. I'm not sure if the special case holds for all real numbers.
