I occasionally write expository notes on both elementary and advanced topics, and I typically share them here once they are polished enough. I'd be more than happy if they help someone gain a better understanding.
Combinatorics
Erdős-Stone-Simonovits theorem
Triangle removal lemma
Stanley sequence
Behrend's construction
Szemerédi-Trotter theorem
Analysis
Fréchet derivative
Inverse function theorem
Implicit function theorem
Upper and lower limits of sequences
Multiples of an irrational number form a dense subset
Upper and lower limits of functions
Criteria for functions that are non-monotone
Algebra
Finitely generated abelian groups
Metric Vector Spaces: The Theory of Bilinear Forms