Construction of number systems beyond the naturals (integers, rationals, reals, complex), fundamental constants, and limit-based concepts. The bridge from discrete arithmetic to continuous mathematics.
analysis
Integers (ℤ)
The set ℤ constructed from naturals by introducing additive inverses: {..., -2, -1, 0, 1, 2, ...}.
Rational numbers (ℚ)
The set ℚ of ratios of integers with nonzero denominator.
Real numbers (ℝ)
The complete ordered field extending the rationals, typically constructed via Dedekind cuts or Cauchy sequences.
Complex numbers (ℂ)
The field ℝ² with multiplication (a,b)(c,d) = (ac-bd, ad+bc); isomorphic to ℝ[x]/(x²+1).
π (pi)
The ratio of a circle's circumference to its diameter in Euclidean geometry. Irrational and transcendental.
e (Euler's number)
The base of the natural logarithm, equal to the limit of (1 + 1/n)^n as n→∞. Irrational and transcendental.
i (imaginary unit)
A symbol i satisfying i² = -1; the foundation of the complex numbers.
φ (golden ratio)
The positive solution to x² = x + 1; ≈ 1.6180339887.
Pythagorean theorem
In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
Limit of a sequence
ε-N definition: lim a_n = L iff ∀ε>0 ∃N s.t. n≥N ⇒ |a_n − L| < ε.
Limit of a function
ε-δ definition: lim_{x→a} f(x) = L iff ∀ε>0 ∃δ>0 s.t. 0<|x-a|<δ ⇒ |f(x)-L|<ε.
Continuity
f is continuous at a iff lim_{x→a} f(x) = f(a). f is continuous on a set if it is continuous at every point.
Cauchy sequence
A sequence (a_n) is Cauchy iff ∀ε>0 ∃N s.t. m,n≥N ⇒ |a_m − a_n| < ε. Encodes convergence without knowing the limit.
Completeness of ℝ
Every Cauchy sequence in ℝ converges. Equivalently: every non-empty bounded set of reals has a supremum.
Bolzano-Weierstrass theorem
Every bounded sequence in ℝⁿ has a convergent subsequence.
Derivative
f'(a) = lim_{h→0} (f(a+h) − f(a))/h. A linear approximation: the best slope that fits f near a.
Mean value theorem
If f is continuous on [a,b] and differentiable on (a,b), then ∃c∈(a,b) with f'(c) = (f(b)−f(a))/(b−a).
Riemann integral
∫_a^b f dx = limit of Riemann sums Σ f(x_i*)Δx_i as the partition is refined, provided the limit exists.
Fundamental theorem of calculus
If F' = f and f is integrable, ∫_a^b f = F(b) − F(a). Differentiation and integration are inverse operations.
Series convergence
A series Σ a_n converges iff its sequence of partial sums S_N = Σ_{n≤N} a_n converges. Tests: comparison, ratio, root, integral,…
Uniform convergence
f_n → f uniformly on S iff sup_{x∈S} |f_n(x) − f(x)| → 0. Stronger than pointwise; preserves continuity and swaps limits with integrals.
Taylor series
Near a, an infinitely differentiable f is represented by Σ f^(n)(a)(x-a)^n/n!, when the series converges to f. Power-series expansion of…
τ (tau = 2π)
The full-turn circle constant, equal to 2π. Many formulas (Fourier, period of sin/cos, circumference-over-radius) are cleaner in terms of τ.
√2 (Pythagoras's constant)
The positive square root of 2; irrational. Length of the diagonal of a unit square; first irrational proven by the Pythagoreans.
√3 (Theodorus's constant)
The positive square root of 3; irrational. Appears as the height of an equilateral triangle of side 2 and in hexagonal geometry.
γ (Euler–Mascheroni constant)
The limiting difference between the harmonic series and the natural logarithm; irrationality is an open problem.
ln 2
The natural logarithm of 2; irrational and transcendental. Appears in entropy of a fair coin (in nats), alternating harmonic series sum.
ζ(2) = π²/6 (Basel constant)
The Basel sum — the sum of reciprocals of squares. Solved by Euler in 1735; equal to π²/6.
ζ(3) (Apéry's constant)
The sum of reciprocals of cubes. Proven irrational by Apéry in 1978; whether it is transcendental is open.
G (Catalan's constant)
Catalan's constant G; appears in combinatorics and evaluating integrals of the form ∫₀¹ dx/(1+x²) · ln-like factors. Irrationality is open.
K (Khinchin's constant)
The geometric mean of the continued-fraction digits of almost every real number converges to K, independently of the number chosen.
α (Feigenbaum α)
The universal scaling ratio of successive bifurcation widths in one-dimensional maps undergoing period-doubling (Feigenbaum's first…
δ (Feigenbaum δ)
The universal ratio of successive bifurcation-parameter intervals in period-doubling cascades. Independent of the specific map family…
ρ (plastic ratio)
The unique real root of x³ = x + 1; algebraic, degree 3. Plays the role for cubics that φ plays for quadratics.
Ω (Omega constant)
The unique real solution of Ω·e^Ω = 1; equals the Lambert-W function evaluated at 1. Appears in solving transcendental equations.
A (Glaisher–Kinkelin constant)
The Glaisher–Kinkelin constant, arising in the asymptotics of the hyperfactorial H(n) = 1^1·2^2·…·n^n and in regularised ζ′(−1).
Laplace limit
The radius of convergence of Laplace's series expansion for the eccentric anomaly in terms of eccentricity (orbital mechanics). Beyond this…
Fransén–Robinson constant
The area under the reciprocal-gamma curve from 0 to ∞; appears as the expected value of a random permutation's displacement statistic and…
Erdős–Borwein constant
Sum of reciprocals of Mersenne numbers. Proven irrational by Erdős (1948); transcendence is open.
K (Sierpiński's constant)
Asymptotic correction term in the average number of representations of an integer as a sum of two squares: Σ_{k≤n} r₂(k)/k = π ln n + K +…
Intermediate value theorem
If f : [a,b] → ℝ is continuous and y is between f(a) and f(b), then ∃c ∈ [a,b] with f(c) = y. Follows from completeness of ℝ.
Extreme value theorem
A continuous function on a compact set attains its supremum and infimum. The real-line case f : [a,b] → ℝ is the classical statement.
Heine–Borel theorem
A subset of ℝ^n is compact iff it is closed and bounded. The prototypical characterisation of compactness in finite-dimensional Euclidean…
Rolle's theorem
If f ∈ C([a,b]) is differentiable on (a,b) and f(a)=f(b), then ∃c ∈ (a,b) with f'(c)=0. The special case of MVT that Lagrange generalised.
Power series
A formal/analytic expression Σ a_n (x−c)^n. Its analytic properties on the disk of convergence ground complex analysis and…
Radius of convergence
For a power series Σ a_n (x−c)^n, there exists R ∈ [0,∞] (Cauchy–Hadamard: 1/R = limsup |a_n|^{1/n}) such that the series converges…
Uniform continuity
f : X → Y is uniformly continuous if ∀ε>0 ∃δ>0 ∀x,y: d(x,y)<δ ⇒ d(f(x),f(y))<ε (same δ works everywhere). Heine: continuous ⇒ uniformly…
Absolute continuity
f : [a,b] → ℝ is absolutely continuous iff ∀ε>0 ∃δ>0 such that for any finite collection of disjoint intervals (x_i,y_i) with Σ(y_i−x_i)<δ,…
Banach space
A complete normed vector space (V, ‖·‖). Hahn–Banach, open mapping, closed graph, and uniform boundedness theorems form the four pillars…
Hilbert space
A complete inner-product space. The natural setting for L² and quantum mechanics; Riesz representation and spectral theorems are key.
Fréchet space
A complete metrisable locally-convex topological vector space whose topology is generated by a countable family of seminorms. Example:…
Schwartz space 𝒮(ℝ^n)
The space of smooth functions f : ℝ^n → ℂ all of whose derivatives decay faster than any polynomial. Fourier transform is a topological…
Distribution (generalised function)
Schwartz's theory: a distribution is a continuous linear functional on the test-function space 𝒟(Ω) (or 𝒮 for tempered distributions). …
Sobolev space W^{k,p}
The Banach space of L^p functions whose weak derivatives up to order k lie in L^p. Essential for variational PDE: embeddings, trace, and…
Stone–Weierstrass theorem
A unital subalgebra A ⊆ C(X) (X compact Hausdorff) that separates points is dense in C(X) w.r.t. the uniform norm. Weierstrass polynomial…
Convex set
A subset C ⊆ ℝⁿ is convex iff for every x, y ∈ C and λ ∈ [0, 1] the point λx + (1−λ)y lies in C. Equivalently, C contains every line…
Spectral theorem (unbounded, self-adjoint)
Self-adjoint operator A on Hilbert space ↔ projection-valued measure + multiplication-operator representation. von Neumann. Foundation of…
Distributions (Schwartz)
Continuous linear functionals on test functions 𝒟(ℝⁿ). Generalised functions: δ, principal value, weak derivatives. Sobolev spaces, PDE.
Rellich-Kondrachov compactness
On bounded Lipschitz Ω ⊂ ℝⁿ: embedding Wᵏ'ᵖ(Ω) ↪↪ Lᵠ(Ω) is compact for 1/q > 1/p − k/n. Cornerstone of variational PDE.
Poincaré inequality
For u ∈ W^{1,p}_0(Ω), Ω bounded: ||u||_p ≤ C(Ω) ||∇u||_p. Basis of variational methods.
Calderón-Zygmund theory
Singular integral operators of convolution type Tf = p.v. ∫ K(x−y)f(y)dy with |K(x)|≤C/|x|ⁿ, ∇K ≤ C/|x|^{n+1}. Lᵖ-boundedness via CZ…
Hausdorff-Young inequality
||f̂||_{p'} ≤ ||f||_p for 1 ≤ p ≤ 2, 1/p + 1/p' = 1. Beckner's sharp constants; Babenko 1961.
Hahn-Banach theorem
Sublinear p on V, bounded linear f on subspace M with f ≤ p on M → extends to F on V with F ≤ p. Foundation of duality theory.
Open mapping theorem
Surjective continuous linear T: X→Y between Banach spaces is open. Consequences: closed graph theorem, bounded inverse theorem.
Banach-Steinhaus (uniform boundedness)
Family {T_α} ⊂ B(X,Y) pointwise-bounded on Banach X ⟹ uniformly bounded in operator norm. Baire category argument.
Riesz representation
Continuous linear functional on Hilbert space H: φ(x) = ⟨x,y⟩ for unique y ∈ H. Dual H* ≅ H. Analogue for C(K)* = Radon measures.
Spectral theorem (compact self-adjoint)
Compact self-adjoint T on H: exists ON basis {eₙ} of eigenvectors, eigenvalues λₙ → 0. Tx = Σ λₙ ⟨x,eₙ⟩ eₙ.
Trace-class / Hilbert-Schmidt
||T||_1 = Σ σₙ(T) (singular values); Hilbert-Schmidt ||T||_2 = (Σ σₙ²)^{1/2}. Ideals in B(H); trace well-defined on S_1. QM density…
Fredholm operator
Bounded T with finite-dim ker, finite-dim coker, closed range. Index ind(T) = dim ker − dim coker is locally constant; Atiyah-Singer.
Riesz-Thorin interpolation
Linear T bounded Lᵖ⁰→Lᵠ⁰ and Lᵖ¹→Lᵠ¹ ⟹ bounded L^{pθ}→L^{qθ} for 1/pθ = (1−θ)/p0 + θ/p1. Convexity of operator norms.
Marcinkiewicz interpolation
Weak-type (pᵢ,qᵢ) interpolation: T bounded from weak-Lᵖⁱ to weak-Lᵠⁱ ⟹ strong bounded L^{pθ}→L^{qθ}. Key for Calderón-Zygmund.
Birkhoff ergodic theorem
T measure-preserving on (X,μ), f ∈ L¹: (1/n) Σ f(Tⁱx) → E[f|ℐ] a.e. Time average = space average when T ergodic.
von Neumann mean ergodic theorem
Isometry U on Hilbert H: (1/n) Σ Uⁿx → P_{fix U} x in norm. L²-version of Birkhoff.
Weak / weak-* convergence
x_n ⇀ x iff φ(x_n) → φ(x) for all φ ∈ X*. Bounded sets weakly pre-compact (reflexive X); Banach-Alaoglu: B_{X*} weak-* compact.
Bochner integral
Integral of Banach-valued functions via simple approximants. Bochner measurable + ∫||f||dμ < ∞. Foundation of vector-valued PDE.
Young's convolution inequality
||f*g||_r ≤ ||f||_p ||g||_q, 1/p + 1/q = 1 + 1/r. Sharp version Brascamp-Lieb 1976.