# 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 Ki 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.

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