Common inequalities – Cauchy–Schwarz, Hölder, Minkowski, Young, Chebyshev, and elementary inequalities, with proofs.
Undergraduate review – limits superior, Fatou’s lemma and monotone convergence, differentiation of monotone and absolutely continuous functions, and Baire category.
Appendix: Functional analysis – the open mapping, bounded inverse, closed graph, uniform boundedness, and Hahn–Banach theorems.