Index

A

a.e. a.e. convergence abelian group absolute continuity of measures absolutely continuous, [1], [2] absolutely continuous function absolutely continuous measure, [1] abstract category AC AC and a.e. zero derivative implies constant AC implies a.e. differentiability AC implies BV AC is equivalent to being an indefinite integral

accumulation point additive adjoint algebra algebra of functions algebra of sets algebraic lattice algebraic structure antichain antisymmetric arity associative algebra Ā

B

Baire category theorem, [1], [2] Banach space, [1], [2], [3], [4] Banach space of bounded linear maps with complete codomain Banach-Steinhaus theorem Banach-Steinhauss principle of uniform boundedness base bilinear algebra binary operation Bolzano-Weierstrass theorem, [1] Boolean algebra homomorphism Borel (-measurable) function

Borel function Borel measurable function Borel measure, [1] Borel set, [1] Borel σ-algebra bounded convergence theorem bounded linear functional bounded linear transformation bounded linear transformations bounded set, [1] bounded variation B̄

C

canonical normal form Caratheodory's theorem Cartesian product category of categories Cauchy sequence chain characteristic function Choice clone closed closed ball closed graph theorem, [1] closed set closure closure operator cocomplete cocomplete poset cod codomain cofinite topology commutative diagram commutative group compact element compact Hausdorff space compact set, [1] compactness theorem complete complete Banach space

complete lattice complete lattice homomorphism complete measure complete measure space complete metric space complete normed linear space complete poset complete set complex measure component composition of operations concentrated concrete category conjugate exponents, [1] consecutive functions continuity theorem continuous continuous and measurable functions continuous function, [1] continuous functions on [-1,1] are dense in L₁[-1,1] continuous linear transformations contravariant powerset functor coproduct countably additive countably subadditive covariant powerset functor cover covering Curry-Howard correspondence

D

dcpo dct dense set density (of continuous functions) derivative of the integral diameter diameter of a set differentiability of increasing functions directed graph

directed set directed-cocomplete poset directed-cocomplete preorder discrete topology division ring dom domain dominated convergence theorem, [1] (version 2) dual

E

Egoroff's theorem, [1], [2], [3], [4] em endofunctor endomorphism epimorphism equivalence class equivalence relation equivalent categories

essentially surjective on objects Euclidean norm, [1] Euclidean space eval evaluation functor evaluation natural transformation even function existential image functor extensional

F

faithful functor Fatou's lemma, [1], [2] field finite measure finite ordinals finite set first category fork free algebra free monoid free object

Fubini's theorem, [1], [2] Tonelli's theorem, [1], [2] function extensionality functional functor functor category Fundamental theorem of calculus f̂ f̃ f̄ fₗ

G

Galois connection Galois pair general composition generalized element

global element graph morphism group groupoid

H

Hahn-Banach theorem, [1] Hausdorff space, [1] height Heine-Borel theorem, [1] Hellinger-Toeplitz theorem Heyting algebra Heyting algebra homomorphism

Hilbert space, [1], [2], [3] hom set homeomorphic homeomorphism homomorphism Hölder's inequality, [1], [2] h₁ h₂

I

idempotent impredicative indiscrete topology inductive set initial object inner product inner product space integrability of a measurable function integrable

integral integral extensionality intensional interior inverse function theorem, [1] inverse mapping theorem invertible linear map isometrically isomorphic isometry, [1] isomorphism

K

kernel

Kleene closure

L

L lambda calculus lattice lattice homomorphism law of the excluded middle Lebesgue integrable Lebesgue integral Lebesgue measurable function Lebesgue measurable set Lebesgue measure, [1], [2] Lebesgue null set Lebesgue outer measure left module Leibniz equal lift (n)

lifts (v) limit point linear functional, [1], [2] linear operator linear space linear transformation Lipschitz condition Lipschitz constant Lipschitz continuous Littlewood's principle locally compact locally small category logically equivalent lower limit Lusin's theorem

M

magma mct mean value theorem (version 2) measurability of upper limit measurable function measurable set measurable space measure measure space metric space

middle linear map Minkowski's inequality, [1] module, [1] monoid monoid homomorphism monomorphism monotone convergence theorem, [1] monotone function monotone increasing function morphism multiplicative inverse mutually singular

N

natural isomorphism natural transformation naturally isomorphic negative part negligible neighborhood

nonnegative function norm normed linear space, [1], [2], [3], [4] normed vector space nowhere dense nullary operation

O

odd function open ball open covering open mapping open mapping theorem open set

operator norm opposite category orthogonal set orthonormal basis orthonormal set outer measure

P

parallel morphisms Parseval's equality partial function partial order partial ordering partially ordered set point point-wise convergence pointwise limit polynomials poset positive measure positive part

power set operator powerset functor preorder preserves principle of uniform boundedness Banach-Steinhauss product product topology product σ-algebra projection operation projection operator properties of measures properties of the Lebesgue integral proposition extensionality

Q

quotient

R

Radon-Nikodym corollary Radon-Nikodym derivative, [1], [2], [3], [4], [5] Radon-Nikodym theorem, [1], [2] (full version) ran reduct reflexive relation relational product relational structure

relative topology respects retract retraction Riesz representation theorem, [1], [2] (version 2) right module ring ring of sets ring with unity

S

second category second countable space section self-adjoint self-dual semigroup seminorm semiring of sets separable separable Hilbert space, [1], [2] separable space separates the points signature

signed measure simple function simplex category small category source vertex step function Stone-Weierstrass theorem, [1] (version 2) subadditive subalgebra subdcpo subuniverse symmetric

T

target vertex terminal object ternary operation Tonelli's theorem Fubini's theorem, [1], [2] topological space topology total function

total order totally bounded totally bounded set transitive translation invariance triangle inequality Tychonoff's theorem type theory

U

unary operation underlying set functor uniform boundedness principle uniform boundedness theorem uniform convergence uniformly continuous, [1] unique morphism property

unit unital ring unitary operator universal image functor universal mapping property universal property universe upper limit

V

valuation

variety vector space

Α

α

Β

β

Γ

Γ

γ

Δ

Δ

δ

Ε

ε

Ι

ι

Κ

κ

Λ

Λ

λ

Π

Π

π

Ρ

ρ

Σ

Σ σ	σ-algebra σ-finite measure

Φ

Φ

φ

Χ

Χ

χ

Ψ

Ψ

ψ

Ω

Ω ω ω-chain

ω-chain cocomplete ω-chain cocomplete poset ω-cpo

Φ

ϕ

ℂ

ℂ

ℓ

ℓ

ℕ

ℕ

ℝ

ℝ

ℝ*

ℤ

ℤ

ℬ

ℬ

ℵ

ℵ

𝒫

𝒫

𝔅

𝔅

𝔐

𝔐

𝔸

𝔸

𝔹

𝔹

𝕋

𝕋

𝚤

𝚤

𝚥

𝚥

Symbols

¬ × ← ↑ → ↓ ↠ ↣ ↦ ∀ ∃ ∅ ∈ ∉ ∋ ∌ ∏ ∑ ∗ ∘ ∞-norm ∧ ∨ ∩ ∪ ∼

≅ ≈ ≡ ≤ ≥ ≪ ≫ ⊂ ⊃ ⊆ ⊇ ⊢ ⊧ ⊨ ⋀ ⋁ ⋂ ⋃ ⋆ ⋈ ⟨ ⟩ ⟹ ⟺ ⤖ ⫢