Alphabeta MathFunctional Analysis
Functional analysis studies infinite-dimensional vector spaces by combining algebraic structure with topology and completeness. The collection begins with normed and Banach spaces, their completions, bounded operators, quotient spaces, and the geometric and analytic forms of Hahn--Banach. It then develops the Baire-category principles behind uniform boundedness, open mapping, and closed graph results, before turning to duality, weak topologies, compactness, and the geometry of Banach and Hilbert spaces. Later pages treat compact and unbounded operators, Banach and -algebras, spectral measures, and the functional calculi that organize operator theory. The final part connects this abstract machinery to Fourier analysis, Schwartz functions, distributions, and tempered distributions. Measure theory supplies the examples and integration framework used throughout, while topology and real analysis supply the metric, compactness, and Baire foundations. The resulting track is intended both as a proof-based introduction and as infrastructure for probability, partial differential equations, harmonic analysis, and mathematical physics.
Pathway
The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.
Part 1 · Normed spaces and completeness
32 pagesBanach spaces: completions, quotients, finite-dimensional equivalence, Riesz's lemma, Baire category, open mapping under DC and uniform boundedness under ; Fourier analysis: convolution, inversion, Schwartz Parseval, Plancherel, Poisson summation, tempered distributions; Hahn--Banach separation and norming, duality, weak and weak-star topologies, locally convex spaces, Banach--Alaoglu, Krein--Milman, reflexivity and James; Schauder bases, Bochner integration and the Radon--Nikodym property; Hilbert geometry, projections and Riesz representation; orthonormal bases and Parseval; compact operators and Fredholm theory; Banach algebras, spectra and the operator calculi; Gelfand theory; the compact self-adjoint spectral theorem with Hilbert--Schmidt kernels and trace-class operators; Fréchet calculus; Stone's theorem; Fredholm determinants and Lidskii's trace formula for trace-class operators; measurable Hilbert fields, direct integrals and the spectral multiplicity model for separable abelian von Neumann algebras.
The Fourier transform here uses the phase e^-2π ix·ξ on complex-valued functions.
1 definition, 6 lemmas, 8 theorems, 3 corollaries, 2 remarksExamples & counterexamples →- Normed and Banach Spaces20 results
This page fixes the basic normed-space vocabulary used later in functional analysis and keeps the route deliberately small.
6 definitions, 7 lemmas, 4 theorems, 1 corollary, 2 remarksExamples & counterexamples → This page packages the first functional-analysis uses of norm completeness: bounded linear and bilinear maps, the operator norm and the Banach space B(X,Y), and quotient norms with their factorization property.
10 definitions, 4 lemmas, 12 theoremsExamples & counterexamples →This page packages the first finite-dimensional rigidity results for normed spaces: coordinate-space models, norm equivalence, completeness, closedness of finite-dimensional…
1 definition, 2 lemmas, 7 theorems, 5 corollaries, 1 remarkExamples & counterexamples →Schwartz space consists of actual smooth functions with bounded polynomially weighted derivatives.
2 definitions, 7 lemmas, 9 theorems, 2 corollariesExamples & counterexamples →A pointwise bounded sequence of bounded linear maps on a Banach space has uniformly bounded norms under AC ω.
1 lemma, 1 theoremExamples & counterexamples →- The Analytic Hahn Banach Theorem13 results
This page follows the analytic route through Hahn-Banach. It starts with sublinear domination, isolates the one-step interval calculation and the chain-union upper bound…
2 definitions, 3 lemmas, 5 theorems, 2 corollaries, 1 remarkExamples & counterexamples → The Baire theorem turns pointwise information into uniform bounds, then supplies the open-mapping and closed-graph principles.
2 definitions, 3 lemmas, 7 theorems, 4 corollariesExamples & counterexamples →Tests are smooth functions with compact support. Their topology controls every derivative while allowing the support to vary through compact subsets of the domain. We…
15 definitions, 12 lemmas, 21 theorems, 1 corollaryExamples & counterexamples →The gauge turns a translated open convex neighbourhood into a real sublinear functional.
6 definitions, 4 lemmas, 6 theorems, 5 corollariesExamples & counterexamples →- Norming and Separation under Hahn–Banach12 results
The real dominated-extension principle HB is stated as an additional assumption over ZF.
3 definitions, 4 lemmas, 3 theorems, 2 corollariesExamples & counterexamples → Continuous linear functionals turn quotient spaces into annihilators and subspaces into dual quotients.
5 definitions, 10 lemmas, 10 theorems, 4 corollaries, 2 remarksExamples & counterexamples →A locally convex space has enough convex zero-neighborhoods to replace norm balls in continuous separation.
3 definitions, 4 lemmas, 3 theoremsExamples & counterexamples →Tempered distributions are the continuous bilinear dual of Schwartz space.
5 definitions, 5 lemmas, 14 theoremsExamples & counterexamples →- Weak and Weak Star Topologies19 results
The weak topology records all bounded scalar measurements of a vector.
4 definitions, 4 lemmas, 9 theorems, 2 corollariesExamples & counterexamples → Weak-star compactness begins with a concrete product embedding of the dual ball.
3 definitions, 2 lemmas, 8 theorems, 4 corollariesExamples & counterexamples →- Reflexivity and Eberlein Smulian28 results
Reflexivity first appears here as a compactness property of the closed unit ball and as a canonical relationship between a Banach space and its bidual.
3 definitions, 10 lemmas, 11 theorems, 4 corollariesExamples & counterexamples → The Bochner integral is built from finite Banach-valued simple functions with the zero-times-infinity convention made explicit and representation independence proved before use.
6 definitions, 11 lemmas, 12 theorems, 1 corollaryExamples & counterexamples →A Schauder basis is an ordered, fixed-expansion structure, not merely a dense sequence.
9 definitions, 11 lemmas, 11 theorems, 3 corollariesExamples & counterexamples →The page begins from the published definition of a bounded linear operator and the metric notion of compactness, and defines a compact operator by the compact closure of the…
4 definitions, 10 lemmas, 10 theorems, 2 corollariesExamples & counterexamples →The pair begins from the published definition of a real or complex inner product and the library's normed-space conventions, and it records the induced length as the published inner-product norm.
6 definitions, 6 lemmas, 10 theorems, 2 corollaries, 2 remarksExamples & counterexamples →This page develops the spectral theory of a single element of a unital complex Banach algebra, from the Neumann series to the holomorphic functional calculus.
11 definitions, 9 lemmas, 10 theorems, 1 corollaryExamples & counterexamples →This page supplies the nonlinear differential calculus that the linear operator theory of the preceding pages does not contain, and it does so in the Fréchet, not the…
7 definitions, 4 lemmas, 1 proposition, 5 theorems, 1 remarkExamples & counterexamples →The pair continues the Hilbert-space geometry of the previous page with orthonormal families on arbitrary index sets.
4 definitions, 7 lemmas, 11 theorems, 2 corollariesExamples & counterexamples →This page develops Gelfand theory for commutative Banach algebras together with the commutative Gelfand–Naimark theorem and the topological dictionaries that accompany it.
10 definitions, 9 lemmas, 18 theoremsExamples & counterexamples →The page begins with the Hilbert–Schmidt definition relative to a supplied Hilbert basis: for a bounded operator T:H→ K the square-sum ∑ e∈ E‖Te‖² is the supremum of its…
1 definition, 1 lemma, 3 theoremsExamples & counterexamples →The page begins with the quantitative core of self-adjoint operator theory: the operator norm is the supremum of its quadratic form ‖T‖=sup ‖x‖=1|⟨ Tx,x⟩|, proved over both…
3 definitions, 7 lemmas, 8 theorems, 3 corollariesExamples & counterexamples →This page builds the continuous functional calculus for bounded self-adjoint and bounded normal operators from the Gelfand theory of the preceding pair and uses it to…
5 definitions, 7 lemmas, 11 theorems, 2 corollaries, 1 remarkExamples & counterexamples →This page builds a local Fredholm determinant theory for trace-class operators on complex Hilbert spaces.
3 definitions, 12 lemmas, 1 proposition, 1 theoremExamples & counterexamples →This page refines the continuous functional calculus of the preceding pair into the measurable calculus of a bounded normal operator.
5 definitions, 6 lemmas, 9 theorems, 1 corollaryExamples & counterexamples →This page builds the measurable bookkeeping that lets a family of Hilbert spaces be integrated along a measure space, together with the operators that respect the family.
4 definitions, 3 lemmas, 4 theoremsExamples & counterexamples →This page extends the bounded PVM calculus of the preceding pair to unbounded self-adjoint operators and ends with Stone's theorem and the min-max principle.
15 definitions, 8 lemmas, 15 theorems, 2 corollaries, 1 counterexampleExamples & counterexamples →