Part VII: Analysis

Analysis is the rigorous theory of limits. Calculus is its application. The entire edifice of physics, probability, and information theory rests on the analytic foundations laid in the 19th century.

7.1 Real Analysis

Sequences: A sequence (aₙ) in ℝ converges to L if for every ε > 0 there exists N such that n > N ⟹ |aₙ − L| < ε.

Series: ∑ aₙ converges if the sequence of partial sums converges.

Convergence tests: Ratio test, root test, comparison test, alternating series test. These determine whether a series converges without computing its limit.

Continuity (ε-δ definition): f: ℝ→ℝ is continuous at c if for every ε > 0 there exists δ > 0 such that |x − c| < δ ⟹ |f(x) − f(c)| < ε.

The fundamental theorems of calculus:

1. If F(x) = ∫_a^x f(t)dt with f continuous, then F'(x) = f(x)
2. ∫_a^b f'(x)dx = f(b) − f(a)

Uniform convergence: A sequence of functions fₙ → f uniformly if the convergence rate is independent of the point. Uniform convergence preserves continuity, integrability, and differentiability (under suitable conditions); pointwise convergence does not.

Power series: ∑ aₙ(x−c)ⁿ converges on an interval (c−R, c+R) where R = 1/limsup |aₙ|^(1/n) is the radius of convergence. Within the radius, the series defines an analytic function.

7.2 Complex Analysis

Holomorphic functions: A function f: ℂ→ℂ is holomorphic at z₀ if the complex derivative f'(z₀) = lim_{h→0} [f(z₀+h) − f(z₀)]/h exists.

Cauchy-Riemann equations: f = u + iv is holomorphic iff ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x.

Cauchy's integral theorem: If f is holomorphic on and inside a simple closed curve γ, then ∮_γ f(z)dz = 0.

Cauchy's integral formula: f(z₀) = (1/2πi) ∮_γ f(z)/(z−z₀) dz. The value of a holomorphic function at any interior point is determined by its values on the boundary.

Power series expansion: Every holomorphic function is analytic — it has a convergent power series expansion around every point. Holomorphic = analytic (unlike in real analysis, where smooth ≠ analytic).

Residue theorem: ∮_γ f(z)dz = 2πi ∑ Res(f, zₖ) where the sum is over poles zₖ inside γ. This converts contour integrals to local residue calculations — a powerful technique for computing definite integrals over the real line.

Riemann mapping theorem: Every simply connected proper open subset of ℂ is biholomorphic to the open unit disk. Conformal geometry of the plane is essentially determined by the disk.

Riemann Hypothesis: The Riemann zeta function ζ(s) = ∑ n⁻ˢ, analytically continued to ℂ, has zeros at s = −2, −4, −6, … (trivial zeros) and at points on the critical strip 0 < Re(s) < 1. The hypothesis states: all non-trivial zeros have Re(s) = 1/2. Status: open. One of the Millennium Prize Problems.

7.3 Functional Analysis

Normed spaces: A vector space V over ℝ or ℂ with a norm ‖·‖: V→ℝ≥0 satisfying: ‖v‖ = 0 ↔ v = 0; ‖αv‖ = |α|‖v‖; ‖v+w‖ ≤ ‖v‖ + ‖w‖.

Banach spaces: Complete normed spaces. ℝⁿ, Lᵖ spaces, the space of bounded continuous functions with sup norm.

Hilbert spaces: Complete inner product spaces — the infinite-dimensional analogs of Euclidean space. L²(X, μ) (square-integrable functions) is the fundamental example. Quantum mechanics lives in Hilbert space.

Bounded linear operators: T: H→H is bounded if ‖T‖ = sup_{‖v‖=1} ‖Tv‖ < ∞. The bounded operators B(H) on a Hilbert space form a C*-algebra.

Spectral theorem (infinite dimensions): For a self-adjoint bounded operator T on a Hilbert space H, there is a projection-valued measure P such that T = ∫ λ dP(λ). The "diagonalization" of infinite-dimensional operators.

Compact operators: T is compact if it maps bounded sets to precompact sets. Compact self-adjoint operators have discrete, countable spectra — their eigenvalues form a sequence converging to 0.

Hahn-Banach theorem: Every bounded linear functional on a subspace of a normed space extends to the whole space. Fundamental for duality theory.

7.4 Measure Theory

Sigma-algebra: A collection Σ of subsets of X closed under complementation and countable unions. The pair (X, Σ) is a measurable space.

Measure: A function μ: Σ→[0,∞] that is countably additive: μ(⋃ₙ Aₙ) = ∑ₙ μ(Aₙ) for disjoint measurable sets Aₙ.

Lebesgue measure: The unique translation-invariant measure on ℝⁿ with μ([0,1]ⁿ) = 1. Assigns measure zero to countable sets and the Cantor set; assigns positive measure to open intervals.

Lebesgue integration: ∫ f dμ is defined for measurable functions f as the limit of integrals of simple functions. Extends Riemann integration and handles limits correctly:

• Monotone Convergence Theorem: If fₙ ↑ f, then ∫ fₙ → ∫ f
• Dominated Convergence Theorem: If fₙ→f and |fₙ| ≤ g with ∫g < ∞, then ∫ fₙ → ∫ f

L^p spaces: Functions f with ∫ |f|^p dμ < ∞, modulo a.e.-equality. (L², the square-integrable functions, is a Hilbert space.)

7.5 Probability Theory (Measure-Theoretic)

Probability space: (Ω, Σ, P) where Ω is the sample space, Σ the sigma-algebra of events, P a probability measure with P(Ω) = 1.

Random variable: A measurable function X: Ω→ℝ. The distribution of X is the measure P∘X⁻¹ on ℝ.

Expected value: E[X] = ∫_Ω X(ω) dP(ω).

Fundamental limit theorems:

• Law of Large Numbers (LLN): If X₁, X₂, … are i.i.d. with mean μ, then (X₁+…+Xₙ)/n → μ almost surely
• Central Limit Theorem (CLT): (X₁+…+Xₙ − nμ)/(σ√n) → N(0,1) in distribution. The Gaussian is the universal attractor under addition of independent random variables

Conditional expectation: E[X|ℱ] is the unique ℱ-measurable random variable Y with ∫_A Y dP = ∫_A X dP for all A ∈ ℱ. This is the Radon-Nikodym derivative dμ_X/dP restricted to ℱ.

Martingales: A sequence (Xₙ, ℱₙ) with E[Xₙ₊₁|ℱₙ] = Xₙ. Models of fair games, random walks, and stock prices. The optional stopping theorem, Doob's maximal inequality, and convergence theorems govern their behavior.

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