Rudin Crib Notes

Brevity has been chosen over accuracy because the whole point is that you should know this stuff already.

Chapter 2: Basic Topology (+ some Ch. 3)

An isolated point of E is in E but not a limit point of it. E is perfect if it is exactly equal to its set of limit points. Equivalently, it is closed and has no isolated points. Ex. 2.44: The Cantor set is perfect.

A compact set is a set for which every open cover has a finite subcover.

Compactness or compact sets have these properties (with made-up names):

• 2.33, space-independence: still compact regardless of what space you consider it part of
• 2.34, closed
• 2.35, hereditary: closed subsets are compact
• 2.36, corollary telescoping descent: a descending sequence of nonempty compact \(K_1 \supseteq K_2 \supseteq K_3 \supseteq \cdots\) contains at least one point
  • 2.36 actually says if the K_i are compact and any finite subcollection has nonempty intersection, then the entire colleciton has nonempty intersection.
  • 3.10(b), shrinking convergence: if the \(K_i\) get infinitely small, they contain exactly one point.
• 2.37, infinite accumulation: infinite subsets have at least one limit point
  • 3.6, sequential compactness: infinite sequences have convergent subsequences
    • (Note from the future: This is the actual name of this property in topology. In general, neither compactness nor sequential compactness imply the other, but they are equivalent in metric spaces.)

Chapter 4: Continuity

• 4.8: A function is continuous iff it inverts open sets to open sets.
  • Corollary: A function is continuous iff it inverts closed sets to closed sets
• 4.9: A continuous function maps compact sets to compact sets.
• 4.19: A continuous function on a compact domain is uniformly continuous.

Chapter 5: Differentiation

• 5.6(b) is the differentiable function with discontinuous derivative.
• 5.12: derivatives attain intermediate values.
• 5.15, Taylor’s Theorem (not exact form): Let \(f\) be \(n\) times differentiable near \(\alpha\). Then \(f\) is equal to the degree-\(n\) Taylor polynomial near \(\alpha\) plus a remainder \(h(x)(x - \alpha)^n\) with \(h\) tending to 0 at \(\alpha\).
  • Rudin states the theorem with the Lagrange form of the remainder: at \(x\) near \(\alpha\), \(f\) is equal to the degree-\(n\) Taylor polynomial after the coefficient of the degree-\(n\) term is fudged into using \(f^{(n)}(\xi_L)\) (instead of \(f^{(n)}(x)\) for some \(\xi_L\) between α and x.
  • Also the theorem’s conditions are tricky about what exactly is required at endpoints of intervals.

Chapter 8: Some Special Functions

• 8.8, algeraic completeness of \(\mathbb{C}\).

  \(f\) has an infimum of magnitude and attains it. Recenter it at that infimum and note that it behaves like its smallest-degree nonconstant monomial nearby, which means we can perturb its value towards 0 if the infimum is nonzero. So the infimum is zero.
• 8.14: A “Lipschitz-continuous-at-a-point” function is approached by its Fourier series.
• 8.15: A \(2\pi\)-periodic continuous function is uniformly approximate-able by trigonometric polynomials.
• 8.16, Parseval’s theorem:
  • the integral of the square of a function’s absolute difference from its Fourier series tends to 0
  • the “dot product” of two functions tends to the dot product of their Fourier series coefficients (with fudge factors due to non-orthonormality)

• 8.19: the gamma function is the only extension of the factorial function with a convex log.