I’m going to begin writing shorter, more coherent notes on certain topics (primarily focused on early graduate-level topics in the three main theoretical mathematics topics). These will be shorter than than the traditional notes that can be found in my Class Notes section, in part because in graduate school, I plan on taking most of my notes with pencil and paper, and feel like it’s better to focus on a shorter writing style.

All the results, proofs, etc. in all documents that follow are not my own; I do not include citations within the specific texts because keeping track of them all would be too difficult.

Algebra

The following are standard texts.

  • Algebra, Chapter 0 (Aluffi)
  • Abstract Algebra (Dummit and Foote)
  • Algebra (Lang)

Conjugation and the Sylow Theorems

We discuss the nuances of the conjugation action in groups, and use it to prove the Sylow theorems. We then use the Sylow theorems to classify the nature of groups of a particular order.

Analysis

The following are standard texts.

  • Real and Complex Analysis (Rudin)
  • Real Analysis (Folland)
  • Real Analysis (Royden and Fitzpatrick)
  • Measure, Integration, and Real Analysis (Axler)
  • Princeton Lectures on Analysis (Stein and Shakarchi)
  • A Course in Functional Analysis (Conway)
  • Sequences and Series in Banach Spaces (Diestel)
  • An Introduction to Banach Space Theory (Megginson)

Inequalities and the $L_p$-Spaces

We introduce some of the most important inequalities that are used frequently in real and functional analysis. These inequalities include Jensen’s inequality and Young’s inequality (concerning convex functions), which are then used to prove Hölder’s inequality and Minkowski’s inequality (concerning $p$-norms). Afterwards, we define the $L_p$-spaces and show that they are complete.

Egorov’s Theorem and Lusin’s Theorem

The mathematician J.E. Littlewood introduced three principles of real analysis: every measurable set is nearly a finite union of intervals, every measurable function is nearly continuous, and every pointwise convergent sequence is nearly uniformly convergent. Here, we prove (ii) and (iii), which are the substance of Lusin’s Theorem and Egorov’s Theorem respectively.

Extreme Points, the Krein–Milman Theorem, and Applications

We discuss extremal structure in locally convex topological vector spaces, as well as a fundamental result in the theory of topological vector spaces: the Krein–Milman theorem. We also use extremal structure to prove the Stone–Weierstrass Theorem and the Banach–Stone theorem.

Three Convergence Theorems

We discuss and prove the three big theorems of real analysis — the Monotone Convergence Theorem, Fatou’s Lemma, and the Dominated Convergence Theorem.

Topology

The following are standard texts.

  • A Taste of Topology (Runde)
  • Topology (Munkres)
  • An Introduction to Algebraic Topology (Rotman)

Urysohn’s Lemma

We detail the construction necessary to prove Urysohn’s Lemma, which completely characterizes normal topological spaces via separation using continuous functions.

Compactness in Topological Spaces

We discuss compactness in topological spaces, including some characterizations and some important structures in topology such as nets and filters.

Miscellaneous