Alphabeta MathArticle items · 01 — A $C^1$ diffeomorphism satisfies the change-of-variables formula for $L^1$ functions
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage
- Every continuous function on a closed nondegenerate rectangle in ℝᵐ is Riemann integrable
- The multidimensional Darboux and tagged-mesh definitions of the Riemann integral agree
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in ℝⁿ
- Dynkin's pi-lambda theorem
- For n at least one, open sets, closed sets, compact sets, open balls, boxes, rational open boxes, and rational half-open boxes generate the Borel sigma-algebra on ℝⁿ
- The indefinite integral of a nonnegative measurable function is a measure
- L(ℝⁿ) is exactly the completion of the restriction of λₙ to the Borel sets
- A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere
- A nonnegative integral over a null set vanishes
- Monotone convergence for the integral
- Additivity of the nonnegative Lebesgue integral
- Integrable real and complex functions, and their integrals
- The Lebesgue integral is linear on L¹(μ)
- exp(z+w)=exp z exp w, and the complex exponential extends the real exponential
- Euler's formula: exp(iθ)=cosθ+i sinθ for every real θ
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- Pythagorean and parity identities for all six trigonometric functions on their natural domains
- Quarter-turn values and shifts by pi/2 and pi
- Pi as twice the smallest positive zero of cosine
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- The modulus of an integral is bounded by the integral of the modulus
- Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree
- Closure properties of measurable functions used by the integral
- Basic identities for a probability measure
- The nonnegative integral agrees with the simple integral on simple functions
- The integral of a nonnegative simple function
- The exponential addition formula exp(x+y)=exp(x)exp(y)
- The real exponential function and the number e by a power series
- Squares of nonzero elements are positive
- Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- Existence of left and right Haar measures
- Haar measure is positive on nonempty open sets and finite on compact sets
- Uniqueness of left Haar measure up to scale
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- The operator norm is 0 on the zero domain and otherwise equals the largest singular value, attained at a right-singular vector
- Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition
- The addition formulas for sine and cosine
- The derivatives of sine and cosine are cosine and minus sine
- Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
- A function differentiable at c is continuous at c
- A function continuous on an interval I whose derivative vanishes at every interior point of I is constant on I; consequently two such functions with the same derivative differ by a constant
- The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral
- Vector-valued functions f : A → ℝᵐ, their limits and continuity, with the dictionary to the metric notions
- The mean value inequality: if f : [a,b] → ℝᵐ is continuous and differentiable on (a,b) with ‖ f'‖₂ ≤ M, then ‖ f(b)-f(a)‖₂ ≤ M(b-a)
- For a ≤ b and f : [a,b] → ℝᵐ integrable when a<b, ‖∫ₐᵇ f‖₂ ≤ ∫ₐᵇ ‖ f‖₂; for a<b, ‖ f‖₂ is integrable
- A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions
- The second fundamental theorem: if G is differentiable on [a,b] with G' = f and f is integrable, then ∫ₐᵇ f = G(b)-G(a)
- Every continuous function on an interval has a primitive; two primitives differ by a constant; and ∫ₐᵇ f = G(b)-G(a) for any primitive G
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it
- Each ‖·‖ₚ is a norm on ℝⁿ, and the induced metrics are exactly d₁, d₂ and d_∞ of the published metric-spaces page
- The p-norms ‖ x‖ₚ for rational p ≥ 1, and ‖ x‖_∞
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- The integral with oriented limits: ∫ₐᵃ f := 0 and ∫_bᵃ f := -∫ₐᵇ f
- For a<c<b: f is integrable on [a,b] if and only if it is integrable on [a,c] and on [c,b], and then ∫ₐᵇ f = ∫ₐᶜ f + ∫_cᵇ f; with the oriented form for arbitrary a,b,c
- The derivative f'(c) = lim_x → c f(x) - f(c)/x - c of f : A → ℝ at a point c ∈ A that is a limit point of A, and differentiability on a set
- If f is continuous on an interval I and |f'| ≤ M at every interior point, then |f(x) - f(y)| ≤ M|x-y| for all x,y ∈ I, so f is Lipschitz with constant M and uniformly continuous on I
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- The absolute value makes ℝ a metric space: d(x,y) = |x-y| is a metric, its open balls are the intervals (x-r, x+r), and it is unbounded
- The ε-δ limit lim_x → c f(x) = L of f : A → ℝ at a limit point c of A
- Basic properties of the absolute value
- Limit point, isolated point, adherent point, derived set, and dense subset of ℝ
- Unbounded linear operators: domain, graph and extension
- Riesz representation for Hilbert spaces
- Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets
- Real and complex inner product spaces, with the inner product linear in the first argument
- Linear map between vector spaces over the same field
- Choice function
- Complex norm and complex normed space
- Banach space
- A bounded bilinear map between normed spaces
- Real and imaginary parts, complex conjugation, and modulus
- Complex series, absolute convergence, complex power series, and radius of convergence
- Integer powers in the complex field
- The complex numbers as ℝ[x]/(x²+1), with the real embedding and imaginary unit i
- ℂ=ℝ[x]/(x²+1) is a field, every element is uniquely a+bi, and every nonzero element has inverse (a-bi)/(a²+b²)
- The factorial n! and the falling factorial n^k̲, defined by recursion in ℕ
- The canonical natural ι(n) = n · 1_F of a field
- Canonical naturals are positive and strictly increasing
- The complex exponential series converges absolutely for every complex argument
- The class L¹(μ) of integrable functions
- The ε-neighbourhood and the punctured ε-neighbourhood of a point of ℝ
- Open subset of ℝ (every point has a neighbourhood inside it), closed subset (complement open), and clopen
- Complete ordered field (least-upper-bound property)
- Finite, countably infinite, countable, uncountable
- The well-ordering principle
- A nonempty set is at most countable iff it is a surjective image of ℕ
- Complete metric space: every Cauchy sequence converges in the space
- A function is a relation f with (a,b) ∈ f and (a,c) ∈ f implying b = c; f : A → B, the value f(a), domain and codomain
- The natural numbers ℕ (von Neumann)
- The Cartesian product A × B := { z ∈ P(P(A ∪ B)) : ∃ a ∈ A ∃ b ∈ B z = (a,b) }
- The power set P(x) = { z : z ⊆ x }
- The Axiom Schema of Separation: for each formula φ, ∀ p̄ ∀ x ∃ y ∀ z (z ∈ y ↔ (z ∈ x ∧ φ(z,p̄)))
- The principle of mathematical induction
- On ℕ the order is membership: m < n ⟺ m ∈ n
- Every nonzero natural number is a successor
- Random elements and real random variables
- The nonnegative Lebesgue integral
- Absolute value in an ordered field
- The standard formulas ⟨ x,y⟩=∑_k<nxₖ yₖ on ℝⁿ and ∑_k<nxₖyₖ on ℂⁿ are inner products
- The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product
- The inner-product norm is definite, homogeneous, and satisfies the triangle inequality
- The vector space M_m × n(F) := F^ m × n of m by n matrices over a field, with entrywise operations
- For n≥1, the determinant over a commutative ring by the Leibniz formula, and |det A| for a real matrix
- Square-summable families on an arbitrary index set and the space ℓ²(I)
- A Hilbert space with a given orthonormal basis is ℓ² of the index set
- Square-summable orthogonal families have norm-convergent finite sums
- Strongly continuous unitary representations, invariant linear subspaces and intertwiners
- Orthogonality and the orthogonal complement
- Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces
- Orthonormal families, complete orthonormal systems and Hilbert bases
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- The spaces B(X,Y) and B(X) of bounded linear operators
- Real and complex inner-product spaces and their induced length
- The induced length is a norm
- The transpose of a bounded operator
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
CorollaryLibrary prerequisite
A C¹ diffeomorphism satisfies the change-of-variables formula for L¹ functions
CorollaryLibrary prerequisite
exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0
CorollaryLibrary prerequisite
Linearity, monotonicity, and the modulus bound for expectation
CorollaryLibrary prerequisite
The expectation of an indicator is the probability of the event
CorollaryLibrary prerequisite
The exponential is positive and satisfies exp(-x)=1/exp(x)
CorollaryLibrary prerequisite
Every finite-dimensional real or complex inner product space has an orthonormal basis
CorollaryLibrary prerequisite
Normalized Haar probability on a compact group
CorollaryLibrary prerequisite
The operator norm is submultiplicative and satisfies ||T^*T|| = ||T||²
CorollaryLibrary prerequisite
Parity and the Pythagorean identity for sine and cosine
CorollaryLibrary prerequisite
If f : [a,b] → ℝᵐ is differentiable with integrable f' then ∫ₐᵇ f' = f(b)-f(a); and a bounded derivative makes f Lipschitz
DefinitionLibrary prerequisite
Adjoint of a densely defined operator
DefinitionLibrary prerequisite
The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V
DefinitionLibrary prerequisite
The Axiom of Choice
DefinitionLibrary prerequisite
A bounded linear operator between normed spaces
DefinitionLibrary prerequisite
C star algebra
DefinitionLibrary prerequisite
The complex exponential by its power series
DefinitionLibrary prerequisite
Conditional expectation given a sigma algebra
DefinitionLibrary prerequisite
Continuity of f : A → ℝ at a point of A and on A: the ε-δ condition, its agreement with lim_x → c f(x) = f(c) at a limit point, and continuity at an isolated point
DefinitionLibrary prerequisite
The Axiom of Countable Choice (AC_ω)
DefinitionLibrary prerequisite
Densely defined, closed and closable operators, and cores
DefinitionLibrary prerequisite
The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
DefinitionLibrary prerequisite
Expectation of a nonnegative or integrable random variable
DefinitionLibrary prerequisite
The Frobenius norm ‖ A‖_F=(∑_i,j|aᵢⱼ|²)^1/2 on real or complex matrices
DefinitionLibrary prerequisite
The Gram matrix G(v₀,…,vᵣ₋₁)=(⟨ vᵢ,vⱼ⟩)_i,j<r and Gram determinant, with empty value 1
DefinitionLibrary prerequisite
Hilbert direct sums of unitary representations
DefinitionLibrary prerequisite
Hilbert–Schmidt operator and Hilbert–Schmidt norm
DefinitionLibrary prerequisite
Hilbert space
DefinitionLibrary prerequisite