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Article items · 04 — Scalar and complex measures from a pvm
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Projection valued measure
- A complex measure is a finite-valued countably additive set function
- Measures on sigma-algebras
- The total variation |nu|(E) from countable measurable partitions
- Real and complex inner-product spaces and their induced length
- The Hilbert-space adjoint of a bounded operator
- Hilbert projections are linear, self-adjoint and contractive
- The Axiom of Countable Choice (AC_ω)
- Unbounded Borel functional calculus: domains, products, spectral mapping
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- Integral of a measurable function against a projection-valued measure
- Infinitesimal generator of a unitary group
- Dominated convergence
- Fatou's lemma
- Symmetric, self-adjoint and essentially self-adjoint operators
- The Axiom of Choice
- The standard smooth step function
- Cᵏ maps and multi-index derivative notation in Euclidean space
- The chain rule, in one line from Carathéodory: if g is differentiable at c and f is differentiable at g(c), then f ∘ g is differentiable at c with (f ∘ g)'(c) = f'(g(c)) g'(c)
- 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
- The support of a function on ℝⁿ and its compactly supported Riemann integral
- Integer-order Sobolev spaces and their norms
- Weak differentiation ignores null-set changes
- Uniqueness of a weak derivative as an almost-everywhere class
- The Lᵖ norm descends to the quotient and makes Lᵖ a normed space for 1 ≤ p ≤ ∞
- Complex Holder, Minkowski, and the quotient norm
- Minkowski's inequality for integrals, including p = ∞
- The Lᵖ trace operator on a bounded C¹ domain
- Bounded Cᵏ domains and boundary charts
- Surface integration on compact C1 hypersurfaces
- The space Lᵖ(μ) as the quotient by null functions
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Hilbert space
- Explicit compactly supported smooth cutoffs
- Bounded restriction and cutoff localisation in Sobolev spaces
- The half-space trace estimate and the half-space trace operator
- Cᵏ boundary flattening preserves local W^k,p
- Ambient smooth restrictions are dense on bounded Cᵏ domains
- Adjoint of a densely defined operator
- The adjoint is well defined, closed, and reverses inclusions
- Bounded borel pvm integral
- Pvm integral is a star homomorphism
- Monotone convergence for the integral
- Densely defined, closed and closable operators, and cores
- The induced length is a norm
- Complex Lp classes and Euclidean test-function conventions
- The notation Hᵏ and the reserved zero-boundary symbol
- Real and complex inner product spaces, with the inner product linear in the first argument
- Zero-boundary Sobolev space as a norm closure
- A closed subspace of a Banach space is Banach
- L² with the integral pairing is a Hilbert space
- Holder's inequality for integrals, including the endpoint cases
- The closure of a nonempty A is {x : d(x,A) = 0}, equals A together with its limit points, and is the smallest closed superset
- Locally integrable functions as regular distributions
- Weak derivative of a locally integrable function
- Distributional differentiation is continuous and commutes
- Leibniz rule for distributions
- Multiplication of a distribution by a smooth function
- Chain rule for globally Lipschitz scalar maps of Sobolev functions
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- Assuming countable choice, Lebesgue outer measure is an outer measure that restricts to elementary volume
- Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable
- Finite, sigma-finite, and semifinite measures
- Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space
- Open ball, closed ball and sphere in a metric space
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it
- The p-norms ‖ x‖ₚ for rational p ≥ 1, and ‖ x‖_∞
- Each ‖·‖ₚ is a norm on ℝⁿ, and the induced metrics are exactly d₁, d₂ and d_∞ of the published metric-spaces page
- The finite and reverse triangle inequalities for a norm; and for n ≥ 1 every norm N on ℝⁿ satisfies N(x) ≤ C‖ x‖₁ and is Lipschitz, hence continuous, for d₂
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
- A compact subset of a metric space is closed and bounded
- Measures are monotone
- Outer measures
- Half-open boxes in ℝⁿ and their volume
- Integer powers aᵐ
- Every complete ordered field is Archimedean
- The nonnegative Lebesgue integral
- The nonnegative integral agrees with the simple integral on simple functions
- The simple integral is monotone, homogeneous, and additive
- Integrable real and complex functions, and their integrals
- Integral over a measurable subset
- Additivity of the nonnegative Lebesgue integral
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- Complete metric space: every Cauchy sequence converges in the space
- Cauchy sequence in a metric space
- The recursion theorem
- For |r| < 1 the sequence rᵏ is null, and for |r| > 1 the sequence |r|ᵏ diverges to +∞
- Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ
- A sequence in a metric space has at most one limit
- Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent
- Metric continuity characterisations, with countable choice for the sequential converse
- Factorisation of bⁿ - aⁿ, and the resulting Lipschitz estimate
- Finite sums and finite products, by recursion
- Laws of finite sums and finite products
- The principle of mathematical induction
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- Inverses of positives are positive, and reciprocation reverses order
- Algebra of limits: sums, scalar multiples, products and quotients
- Laws of integer exponents
- Nonnegativity of a metric is a consequence of the other axioms, not an axiom
- Limits and Cauchy sequences of reals
- The rationals embed densely in the reals
- Real powers of the Japanese bracket act on Schwartz space
- Weighted Fourier transforms of Schwartz functions are dense in L2
- Real-order Bessel-potential completion Hˢ
- Complex completeness, density, and inner product: the consumer interface
- Schwartz space is dense in L2
- Schwartz space and its seminorms
- Tempered distribution
- Weak and strong topologies on tempered distributions
- Fourier transform is a topological automorphism of tempered distributions
- Fourier transform agrees with l one and plancherel transforms
- Polynomial growth functions define tempered distributions
- Bochner-integrable function
- Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable
- A bounded linear operator between normed spaces
- The Banach-valued simple integral is well defined
- The published Riemann change-of-variables theorem already gives the Lebesgue formula for continuous compactly supported integrands
- A C¹ diffeomorphism maps Lebesgue null sets to Lebesgue null sets
- A C¹ diffeomorphism maps Lebesgue measurable sets to Lebesgue measurable sets
- Every nonnegative measurable function admits an explicit increasing sequence of simple approximations
- Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system
- L(ℝⁿ) is exactly the completion of the restriction of λₙ to the Borel sets
- The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- The Cauchy-Schwarz inequality for finite sums
- Minkowski's inequality for finite sums (rational exponent)
- Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0}
- Squaring is monotone on the nonnegatives
- Basic properties of the absolute value
- Absolute value in an ordered field
- The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- Existence and uniqueness of n-th roots: a unique a^1/n ≥ 0 with (a^1/n)ⁿ = a
- Linear independence: a finite list v : n → V is independent when ∑_i<n λᵢ vᵢ = 0_V forces every λᵢ = 0_F, and a subset S ⊆ V is independent when every injective finite list into S is independent
- The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder
- Total differentiability gives a local O(‖h‖₂) increment bound and therefore continuity
- Every Euclidean linear map has a unique matrix and satisfies ‖Lh‖₂≤ K‖h‖₂ for some K≥0
- A linear map L:ℝᵐ→ℝⁿ in Euclidean coordinates
- Fréchet derivative between Banach spaces
- The Fréchet derivative is unique
- A bounded bilinear map between normed spaces
- For a bilinear map, boundedness is equivalent to joint continuity
- Composition satisfies ‖ST‖≤‖S‖ ‖T‖
- 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
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Every natural-number-indexed list of nonempty sets has a choice function on its family of values
- Choice for pairs and countable finite choice
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
LemmaLibrary prerequisite
Scalar and complex measures from a pvm
LemmaLibrary prerequisite
A self-adjoint operator generates a strongly continuous unitary group
LemmaLibrary prerequisite
A smooth bump between concentric Euclidean balls
LemmaLibrary prerequisite
The Sobolev norm descends to equivalence classes
LemmaLibrary prerequisite
The trace agrees with classical restriction for continuous Sobolev functions
LemmaLibrary prerequisite
Spectral form domain and core of a semibounded operator
LemmaLibrary prerequisite
Test function cutoffs and euclidean localization
LemmaLibrary prerequisite
The trace commutes with smooth cutoffs and is chart local
LemmaLibrary prerequisite
The unbounded PVM integral is densely defined, closed and normal
LemmaLibrary prerequisite
The Sobolev space H¹ is a Hilbert space
LemmaLibrary prerequisite
Linearity, locality, and commutation of weak derivatives
LemmaLibrary prerequisite
Weak Leibniz rule with a smooth factor
LemmaLibrary prerequisite
Weak product rule for bounded Sobolev functions
PropositionLibrary prerequisite
Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure
PropositionLibrary prerequisite
Monotonicity and nonnegative homogeneity of the nonnegative integral
TheoremLibrary prerequisite
Absolute continuity of the integral
TheoremLibrary prerequisite
A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point
TheoremLibrary prerequisite
The Bessel completion embeds canonically in tempered distributions
TheoremLibrary prerequisite
Weighted tempered-distribution characterization of Hˢ
TheoremLibrary prerequisite
Bochner integrability criterion
TheoremLibrary prerequisite
Bounded linear maps commute with Bochner integration
TheoremLibrary prerequisite
A C¹ diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions
TheoremLibrary prerequisite
Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
TheoremLibrary prerequisite
Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
TheoremLibrary prerequisite
Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
TheoremLibrary prerequisite
The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
TheoremLibrary prerequisite
Chain sum product and composition rules for Banach derivatives
TheoremLibrary prerequisite