Welcome to the reference page of A Course in Real Analysis.
About
This page is intended to hold only a dense collection of facts with no explanation —
i.e., it is not intended for teaching or learning, but only for reference.
The pedagogical portion of the course is hosted at this GitHub repository:
A Course in Real Analysis ... in Marimo!.
Unit 1: Real Number Axioms
Lessons
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Lesson 2: Groups
- Notation: $xy$, $e$, $x^{-1}$, $x/y$, $x^n$
- Definitions: identity element, inverse element, group, commutative
- Theorems:
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Lesson 3: Rings and Fields
- Notation: $0$, $1$, $na$, $a^n$
- Definitions: distributes, ring, field
- Theorems:
- Lesson 4: Order
- Notation: $\preceq,\prec,\succeq,\succ$
- Defs: Partial order, partially ordered set, poset, commensurability, totality, total order, totally ordered set
- Thm: Poset prec. strict part. ord.
- Thm: Poset neg. prec.
- Lesson 5: Bounds
- Notation: $UB_A, \max(A), \sup(A)$
- Defs: Upper bound, maximum, supremum, and duals (lower bound, minimum, infimum)
- Thm: Bound max. is sup.
- Thm: Bound tot. ord. finite set max.
- Thm: Bound inf. lefteq sup.
- Thm: Bound subset sup. ineq.
- Lesson 6: Function Bounds
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Notation: $\text{Im}(f), \max(f),\sup(f),f\preceq g, f_{(\cdot, c)}, f_{(c,\cdot)}, \sup_{x\in X}f_{(x,\cdot)}, \sup_{y\in Y}f_{(\cdot,y)}, \sup_{x}\sup_y f$
- Defs: Image, function bound, function maximum, function supremum, partially applied function
- Thm: Fun. Bound Ineq. Sup. Ineq.
- Thm: Fun. Bound Double Sup.
- Lesson 7: Ordered Fields
- Notation: $(a,b), (a,b], [a,b), [a,b],\infty, F^\ast, a+X, aX X+Y, XY, X^+, X^-, X^{\preceq 0}, X^{\succeq 0}$
- Def: Interval, bounded, closed, open, positive, negative, compatible with addition, compatible with multiplication
- Thm:
- Thm:
- Thm:
- Lesson 8: Completeness