Alphabeta MathPersonal research
Article items · 06 — Locally integrable functions embed in distributions
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Regular distribution from a locally integrable function
- Local finite order characterization of distributions
- The mollifier family generated by a unit-mass smooth bump
- Every L¹ approximate identity converges to the identity in Lᵖ for 1 ≤ p < ∞
- Dominated convergence
- Test function cutoffs and euclidean localization
- Weak and strong topologies on distributions
- The Axiom of Countable Choice (AC_ω)
- A unit-mass smooth bump generates an L¹ approximate identity
- The half-space trace estimate and the half-space trace operator
- Cᵏ boundary flattening preserves local W^k,p
- Bounded Cᵏ domains and boundary charts
- Bounded C1 domains and their outward normals
- Surface integration on compact C1 hypersurfaces
- Chart and partition independence of surface measure
- Finite ambient partitions near compact sets
- Bounded linear maps extend uniquely across the completion
- Weak Leibniz rule with a smooth factor
- Integer-order Sobolev spaces and their norms
- The space Lᵖ(μ) as the quotient by null functions
- The Axiom of Choice
- A compact metric space is complete and totally bounded, and neither implication uses any choice principle
- In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle
- A sequentially compact metric space is complete, with no choice principle used
- A sequentially compact metric space is totally bounded, proved from the axiom of dependent choice
- A complete, totally bounded metric space is compact, proved from countable choice used exactly once
- Open cover, subcover, compact metric space, and compact subset of a metric space
- Countably compact, sequentially compact and limit point compact metric spaces
- Finite ε-net and totally bounded metric space
- Complete metric space: every Cauchy sequence converges in the space
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- 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
- Order is preserved by adding a constant and by adding inequalities
- Nonnegativity of a metric is a consequence of the other axioms, not an axiom
- Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed
- The multiplicative identity is positive
- Inverses of positives are positive, and reciprocation reverses order
- Sign rules for products and monotonicity of multiplication
- Field
- Ordered field
- Complete ordered field (least-upper-bound property)
- Holder's inequality for integrals, including the endpoint cases
- The function space Lᵖ(μ) for 0 < p < ∞
- The space L^∞(μ) of essentially bounded measurable functions
- The essential supremum is attained as the least essential bound
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Additivity of the nonnegative Lebesgue integral
- Holder's inequality for finite sums and conjugate real exponents
- The Cauchy-sequence reals have the least-upper-bound property
- The reals form a totally ordered field
- Every complete ordered field is Archimedean
- The nonnegative Lebesgue integral
- Integral over a measurable subset
- The indefinite integral of a nonnegative simple function is a measure
- Every nonnegative measurable function is the increasing limit of simple measurable functions
- Continuity from below for measures
- The nonnegative integral agrees with the simple integral on simple functions
- The simple integral is monotone, homogeneous, and additive
- Measure-null sets and almost-everywhere statements relative to a measure
- The simple integral is independent of the chosen representation
- The points polygonally reachable from a fixed point form a clopen subset of every open subset of ℝⁿ
- Every path-connected space is connected, and every path component lies inside a component
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Polygonal paths and polygonally connected subsets of ℝⁿ
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Fermat's interior extremum theorem: if f has a local extremum at a point c interior to its domain and is differentiable at c, then f'(c) = 0
- Local and strict local extrema for scalar fields on Euclidean open sets
- Banach space
- Strongly measurable Banach-valued function
- Finite and countable subadditivity of measures
- Measures are monotone
- Series in the nonnegative extended real line
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Parseval pairing on Schwartz space
- Schwartz space is dense in L2
- Fourier transform is a topological automorphism of Schwartz space
- A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm
- Complex completeness, density, and inner product: the consumer interface
- Continuous functional calculus properties
- Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense
- Commutative Gelfand Naimark
- Spectral permanence for unital c star subalgebras
- Self-adjoint positive unitary and normal elements
- Continuous functional calculus for bounded self adjoint operators
- Spectrum of a positive operator is nonnegative
- Self-adjoint, positive, unitary and normal operators
- C star algebra
- Characters on a unital commutative C star algebra preserve star
- C star algebra generated by a normal operator
- Order on bounded self adjoint operators
- Hilbert-adjoint identities
- The quotient seminorm ‖x+M‖_X/M=inf_m∈ M‖x+m‖=dist(x,M)
- The quotient seminorm is a norm exactly when the subspace is closed
- Series criterion for Banach spaces
- The Lᵖ norm descends to the quotient and makes Lᵖ a normed space for 1 ≤ p ≤ ∞
- Convergence almost everywhere relative to a measure
- Closure properties of measurable functions used by the integral
- Limit superior and limit inferior of a real sequence as infₙ sup_k ≥ n xₖ and supₙ inf_k ≥ n xₖ in ℝ̄
- A real sequence converges to L ∈ ℝ iff liminf xₖ = limsup xₖ = L, and diverges to ±∞ iff both equal ±∞
- Threshold characterisations of real-valued and extended-real-valued measurability
- The half-space trace lies in the fractional Slobodeckij space
- A bounded right inverse of the half-space trace by normal mollification
- The fractional Sobolev space on a compact C¹ boundary
- Chart independence of the fractional boundary norm
- The trace commutes with smooth cutoffs and is chart local
- The coordinate-direction form of the Slobodeckij seminorm
- Bounded Cᵏ domains admit integer-order Sobolev extension
- Interior mollification commutes with weak derivatives
- Complex translation, convolution, approximate identities, and mollification
- Bounded restriction and cutoff localisation in Sobolev spaces
- A smooth bump between concentric Euclidean balls
- Divergence on a bounded C1 Euclidean domain
- Classical normal derivative
- The trace agrees with classical restriction for continuous Sobolev functions
- The Sobolev norm descends to equivalence classes
- Complex Lp completeness and almost-everywhere subsequences
- Complex Holder, Minkowski, and the quotient norm
- Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure
- AC supplies the countable and dependent choices used in Banach integration
- Integral of a measurable function against a projection-valued measure
- The unbounded PVM integral is densely defined, closed and normal
- Cayley correspondence between self-adjoint operators and unitaries
- Cayley transform of a self-adjoint operator
- Spectral theorem for bounded normal operators pvm form
- Support and uniqueness of the spectral measure
- Projection valued measure
- Regular Borel measure on an LCH space
- Spectrum and resolvent of a bounded operator
- Neumann series
- Symmetric, self-adjoint and essentially self-adjoint operators
- Bounded borel pvm integral
- A nonnegative integral over a null set vanishes
- The Lebesgue integral is linear on L¹(μ)
- Integrable real and complex functions, and their integrals
- The completed product measure
- Fubini's theorem for L¹ functions on a sigma-finite product
- A function measurable for a completion is almost everywhere equal to one measurable for the original sigma-algebra
- On a complete measure space, equality almost everywhere preserves measurability
- The completion domain and proposed completed set function of a measure space
- For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique
- Every section of a product-measurable function is measurable
- For sigma-finite measures, the two section-measure integrals of a measurable set agree
- Every nonnegative measurable function admits an explicit increasing sequence of simple approximations
- The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder
- Directional derivatives and partial derivatives of a map U⊆ℝᵐ→ℝⁿ
- The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
- Every Euclidean linear map has a unique matrix and satisfies ‖Lh‖₂≤ K‖h‖₂ for some K≥0
- Ultrafilter
- Zorn's lemma
- The union of a nonempty chain of filters is a filter
- Filter on a set
- Maximal element and greatest element
- Partial order and partially ordered set
- Chain in a poset
- Upper bound, least upper bound, and strict upper bound
- Resolvent and spectrum of an unbounded operator
- Orthogonality and the orthogonal complement
- Pvm integral is a star homomorphism
- Scalar and complex measures from a pvm
- ℚ is countably infinite
- Both ℚ and ℝ ∖ ℚ are dense in ℝ, and every nonempty open subset of ℝ is uncountable
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
- Laplace transform formula for the resolvent
- Exponential bound for a C0-semigroup
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- Classical, strong and mild abstract Cauchy solutions
- The variation-of-constants integral is continuous for integrable forcing
- The generator commutes with the semigroup on its domain
- Time integrals of semigroup orbits lie in the generator domain
- Fundamental theorem of calculus for Banach-valued continuous curves
- Linearity of the Bochner integral
- Bochner integral norm inequality
- Bochner-integrable function
- The generator is closed and densely defined
- Average convergence for a continuous Banach-valued function
- Infinitesimal generator of a C0-semigroup
- Weak convergence of nets and sequences
- Relative Hahn–Banach makes the canonical bidual map an isometry
- If Y is Banach then B(X,Y) is Banach
- Sequential uniform boundedness under countable choice
- The real dominated-extension principle as an additional hypothesis over ZF
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
TheoremLibrary prerequisite
Locally integrable functions embed in distributions
TheoremLibrary prerequisite
The Lᵖ trace operator on a bounded C¹ domain
TheoremLibrary prerequisite
For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice
TheoremLibrary prerequisite
Distinct points of a metric space have disjoint balls around them
TheoremLibrary prerequisite
Minkowski's inequality for integrals, including p = ∞
TheoremLibrary prerequisite
Monotone convergence for the integral
TheoremLibrary prerequisite
A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere
TheoremLibrary prerequisite
For an open subset of ℝⁿ, connectedness, path-connectedness and polygonal connectedness are equivalent
TheoremLibrary prerequisite
A constrained local extremum annihilates every velocity of a differentiable parametrization
TheoremLibrary prerequisite
Bochner Lp is complete for a Banach value space
TheoremLibrary prerequisite
Plancherel theorem
TheoremLibrary prerequisite
Positive square root
TheoremLibrary prerequisite
A quotient of a Banach space by a closed subspace is Banach
TheoremLibrary prerequisite
Riesz-Fischer completeness of Lᵖ for 1 ≤ p ≤ ∞
TheoremLibrary prerequisite
Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable
TheoremLibrary prerequisite
The sharp trace theorem: boundedness and range in the fractional space
TheoremLibrary prerequisite
Ambient smooth restrictions are dense on bounded Cᵏ domains
TheoremLibrary prerequisite
The Gauss-Green integration-by-parts formula with Sobolev traces
TheoremLibrary prerequisite
Integer-order Sobolev spaces are Banach
TheoremLibrary prerequisite
Spectral theorem for unbounded self-adjoint operators (PVM form)
TheoremLibrary prerequisite
Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree
TheoremLibrary prerequisite
Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
TheoremLibrary prerequisite
Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
TheoremLibrary prerequisite
A total derivative computes every directional derivative, and its matrix is the Jacobian
TheoremLibrary prerequisite
The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter
TheoremLibrary prerequisite
Unbounded Borel functional calculus: domains, products, spectral mapping
TheoremLibrary prerequisite
Variation of constants for the inhomogeneous abstract Cauchy problem
TheoremLibrary prerequisite