Alphabeta MathOrthonormal Bases, Parseval and Fourier Series
About this page
Pipeline-generated
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Functional Analysis
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
- Complex Lp Spaces and Test-Function Conventions
- Density Separability and Convolution in Lᵖ
- Lebesgue Measure on Euclidean Space
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Modes of Convergence Egorov and Lusin
- Outer Measure and the Caratheodory Extension Theorem
- Pointwise Extended-Real Function Products
- Product Measures and the Fubini Tonelli Theorems
- Sigma Algebras and Borel Sets
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Areas of Elementary Plane Figures
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Convergence: Nets and Filters
- Countability Axioms and Cardinal Functions
- Function Space Topologies and the Exponential Law
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Separation Axioms: the Hierarchy
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Orthonormal families, complete orthonormal systems and Hilbert bases
Lemma
The finite Bessel inequality and best approximation by a finite orthonormal family
Definition
Square-summable families on an arbitrary index set and the space ℓ²(I)
Theorem
The Bessel inequality for an arbitrary orthonormal family
Lemma
Only countably many coefficients of a square-summable family are nonzero
Lemma
Square-summable orthogonal families have norm-convergent finite sums
Theorem
Parseval equivalences for an orthonormal family
Theorem
Fourier expansion in a Hilbert space
Theorem
Existence of a maximal orthonormal family, and maximality as completeness
Theorem
A Hilbert space with a given orthonormal basis is ℓ² of the index set
Theorem
A Hilbert space with a dense sequence has a finite or countable orthonormal basis
Corollary
A separable infinite-dimensional Hilbert space is ℓ²
Lemma
L² with the integral pairing is a Hilbert space
Definition
The one-dimensional torus and its normalized Haar integral
Lemma
Finite tori are compact Hausdorff spaces separated by characters
Definition
Fourier coefficients and trigonometric polynomials on the torus
Lemma
The trigonometric characters are orthonormal in L² of the torus
Corollary
Trigonometric polynomials are uniformly dense in continuous functions on the torus
Lemma
Continuous functions are dense in Lᵖ of finite tori and of bounded intervals
Theorem
The trigonometric system is complete in L² of the torus
Theorem
Fourier series converge in mean square
Theorem
The Parseval identity for Fourier series
Theorem
Riesz–Fischer: the Fourier coefficient map is onto the space of square-summable families
Theorem
The Fourier basis and Parseval's identity on the finite torus
Sources
- Theo Bühler and Dietmar Salamon, Functional Analysis — Definition 2.62 and Exercise 2.63, p.87
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, pp.47–48
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, pp.47–48, Lemma 2.1
- Theo Bühler and Dietmar Salamon, Functional Analysis — Theorem 2.65 area, p.87
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, pp.48–49 and p.52
- Theo Bühler and Dietmar Salamon, Functional Analysis — Exercise 2.64, p.87
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, p.48, equation (2.4)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, p.48, threshold argument after (2.4)
- Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, printed pp.72–80
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, p.49, Theorem 2.2
- Bruce Blackadar, Ilijas Farah and Asaf Karagila, Hilbert spaces without the Countable Axiom of Choice, §§3–4.1
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, pp.50–51, Theorem 2.4
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, pp.49–50, Theorem 2.2
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, p.52, Theorem 2.7
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, p.52, discussion following Theorem 2.7
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, pp.49–50, Gram–Schmidt and Theorem 2.3
- Theo Bühler and Dietmar Salamon, Functional Analysis — Exercise 2.63, p.87
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.1, p.52, Theorem 2.6
- Theo Bühler and Dietmar Salamon, Functional Analysis — §§1.3.3 and 5.3.1, pp.38–41 and 235–237
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.5, pp.63–64
- Theo Bühler and Dietmar Salamon, Functional Analysis — Example 2.66, pp.87–88
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.5, pp.63–64, equations (2.42)–(2.44)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.5, p.68, Problem 2.19
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.5, p.68, Problem 2.18
- Theo Bühler and Dietmar Salamon, Functional Analysis — §2.3.6, pp.87–88
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.5, pp.64–65, Theorem 2.17
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.5, Theorem 2.17 and equation (2.48)
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.5, Theorem 2.17, final assertion
- Theo Bühler and Dietmar Salamon, Functional Analysis — §§2.3.6 and 5.3.1, pp.87–88 and 235–237
- Gerald Teschl, Topics in Real and Functional Analysis, version November 17, 2017 — §2.5, pp.63–68