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Article items · 07 — Young's inequality for conjugate real exponents
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites and dependency levels 0–1.
1 · Prerequisites
- The weighted arithmetic-geometric mean inequality for real weights
- Real powers for positive bases, with the zero-base positive-exponent convention
- The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents
- The exponential is positive and satisfies exp(-x)=1/exp(x)
- Regular distribution from a locally integrable function
- Test function lf topology universal property
- The Axiom of Choice
- The Axiom of Countable Choice (AC_ω)
- Hilbert space
- Infinitesimal generator of a unitary group
- Projection valued measure
- Strongly continuous one-parameter unitary group
- Symmetric, self-adjoint and essentially self-adjoint operators
- Integral of a measurable function against a projection-valued measure
- A self-adjoint operator generates a strongly continuous unitary group
- Bounded borel pvm integral
- Pvm integral is a star homomorphism
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- Unbounded Borel functional calculus: domains, products, spectral mapping
- Bochner-integrable function
- Strongly measurable Banach-valued function
- ‖τ_h f - f‖ₚ → 0 in Lᵖ(ℝⁿ) as h → 0, for 1 ≤ p < ∞
- Holder's inequality for integrals, including the endpoint cases
- Minkowski's inequality for integrals, including p = ∞
- For a Lebesgue measurable set and every positive ε there is an open superset whose difference from it has outer measure below ε
- L(ℝⁿ) is exactly the completion of the restriction of λₙ to the Borel sets
- Bochner integral norm inequality
- Minkowski's integral inequality
- Continuity from below for measures
- Dominated convergence
- A smooth bump between concentric Euclidean balls
- The support of a function on ℝⁿ and its compactly supported Riemann integral
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Three-dimensional radial Carleman inequality with integer weight
- Weak Leibniz rule with a smooth factor
- Weak derivative of a locally integrable function
- Integer-order Sobolev spaces and their norms
- The essential supremum of a measurable function with respect to a measure
- The space L^∞(μ) of essentially bounded measurable functions
- The essential supremum is attained as the least essential bound
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Cᵏ Euclidean maps and diffeomorphisms
- ℚⁿ is a countable dense subset of ℝⁿ, and rational open boxes form a countable basis
- For n ≥ 1 the product topology on n copies of the usual topology of ℝ is the metric topology of d_∞ on ℝⁿ, and hence also of d₁ and d₂, so ℝⁿ as a product and ℝⁿ as a metric space are one space
- For n ≥ 1 all norms on ℝⁿ are equivalent
- Measure-null sets and almost-everywhere statements relative to a measure
- Finite and countable subadditivity of measures
- 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
- Open ball, closed ball and sphere in a metric space
- Banach space
- Monotone convergence for the integral
- Measures are monotone
- Series in the nonnegative extended real line
- Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable
- The Axiom of Countable Choice (AC_ω)
- Time translations and mollification in Bochner Lp
- Realification preserves a complex norm and its metric
- Bochner-integrable function
- Bochner integrability criterion
- Bochner integral norm inequality
- Bounded linear maps commute with Bochner integration
- Test function space d of an open set
- Distributional derivative
- Banach space
- Strongly measurable Banach-valued function
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- A bounded linear operator between normed spaces
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- Fixed support test function frechet space
- Test function topology
- Linearity of the Bochner integral
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Complex norm and complex normed space
- Distribution
- Integer-order Sobolev spaces and their norms
- Zero-boundary Sobolev space as a norm closure
- The Sobolev norm descends to equivalence classes
- Complex Lp classes and Euclidean test-function conventions
- Complex Holder, Minkowski, and the quotient norm
- Minkowski's inequality for integrals, including p = ∞
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Complex Lp completeness and almost-everywhere subsequences
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- A continuous real function on a compact subset of ℝ is bounded
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure
- The completed product measure
- ℚ is countably infinite
- Both ℚ and ℝ ∖ ℚ are dense in ℝ, and every nonempty open subset of ℝ is uncountable
- Finite and countable subadditivity of measures
- Arithmetic and lattice operations preserve measurability whenever they are defined
- The dual space X^* of a normed space and its dual norm
- Weak topology on a normed space
- Weak convergence of nets and sequences
- The Axiom of Choice
- The trace of a one-dimensional Sobolev function is the pair of endpoint values
- The Axiom of Choice implies countable choice
- The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- Unbounded Borel functional calculus: domains, products, spectral mapping
- Spectral form domain and core of a semibounded operator
- Completion of a normed space
- The metric completion of a normed space carries a unique compatible Banach-space structure
- The norm completion of an inner-product space is a Hilbert space
- A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm
- Hilbert space
- Symmetric, self-adjoint and essentially self-adjoint operators
- Projection valued measure
- A closed subspace of a Banach space is Banach
- Measure spaces
- Topological vector spaces over the real and complex fields
- Finite products of Banach spaces are Banach
- Continuity of a map of topological spaces at a point and globally
- Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not
- 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
- Strongly continuous semigroup
- Bochner dominated convergence theorem
- Bounded linear maps commute with Bochner integration
- For every ε > 0 in a complete ordered field there is a natural n ≥ 1 with 1/n < ε
- The Dirichlet Laplacian generates an analytic heat semigroup
- Semibounded closed-form representation and perturbation
- Integer-order Sobolev spaces are Banach
- Smooth compactly supported functions of an open set are dense in L²
- L² with the integral pairing is a Hilbert space
- Weak derivatives are represented distributional derivatives
- AC supplies the countable and dependent choices used in Banach integration
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- Closed sectorial form and its associated operator
- The unbounded PVM integral is densely defined, closed and normal
- Weak product rule for bounded Sobolev functions
- Chain rule for globally Lipschitz scalar maps of Sobolev functions
- Assuming Countable Choice, Lᵖ-convergent sequences have almost-everywhere convergent subsequences
- Dominated convergence
- Explicit compactly supported smooth cutoffs
- Bounded Cᵏ domains and boundary charts
- Bounded Cᵏ domains admit integer-order Sobolev extension
- The Sobolev space W^k,p is an algebra above the critical index
- Higher-order Sobolev embedding
- Test function cutoffs and euclidean localization
- Weak Leibniz rule with a smooth factor
- The Lᵖ trace operator on a bounded C¹ domain
- Surface integration on compact C1 hypersurfaces
- Chart and partition independence of surface measure
- The space Lᵖ(μ) as the quotient by null functions
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- The notation Hᵏ and the reserved zero-boundary symbol
- Local weak solutions of a divergence-form operator
- Real analytic germs in several variables
- Interior Hᵏ⁺² elliptic regularity
- A product of connected spaces is connected in the product topology, and that argument is a theorem of ZF; for an infinite index set it is the assertion that the product of nonempty spaces is nonempty that uses the Axiom of Choice
- Harmonic functions are real analytic
- Unique continuation for harmonic functions
- The Laplacian of a C² function and of a C² vector field
- A subset of ℝ is connected if and only if it is order-convex, that is, an interval
- Classical derivatives agree with weak derivatives
- The real exponential function and the number e by a power series
- The exponential function is smooth and (exp)'=exp
- The derivatives of sine and cosine are cosine and minus sine
- 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
- Cᵏ Euclidean maps are closed under componentwise algebra and composition
- The Hilbert-space adjoint of a bounded operator
- Hilbert-adjoint identities
- The Lax--Milgram theorem
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Normed vector space over an absolutely valued field
- The reverse triangle inequality in a normed space
- Continuous functions on a closed interval attain their extrema
- Vector addition and scalar multiplication are continuous in a normed space
- Continuity of a map between metric spaces, at a point and globally, in the ε-δ form
- Realification of a complex vector space by restriction of scalars
- Real and complex inner product spaces, with the inner product linear in the first argument
- The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product
- Riesz representation for Hilbert spaces
- The essential supremum is attained as the least essential bound
- The essential supremum of a measurable function with respect to a measure
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Additivity of the nonnegative Lebesgue integral
- A nonnegative measurable function has integral 0 exactly when it vanishes almost everywhere
- Cotangent space and cotangent bundle as a disjoint union
- Geometric control condition for the wave equation with a time-dependent observation domain · Section 1.3.1, Definitions 1.3–1.4 and Proposition 1.5, PDF pp. 6–9; Section 1.3.2, Definition 1.6, PDF p. 9
- On nonlinear Schrödinger equations in exterior domains · Introduction, printed pp. 295–296, opening paragraph
- Star-shaped open subsets of Euclidean space
- Bounded C1 domains and their outward normals
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- Open ball, closed ball and sphere in a metric space
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- A point lies in the closure of A iff every basic neighbourhood of it meets A; the closure is the smallest closed superset and equals A together with its derived set
- The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder
- A total derivative computes every directional derivative, and its matrix is the Jacobian
- In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones
- Distinct points of a metric space have disjoint balls around them
- 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
- The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset
- Fundamental theorem of calculus for absolutely continuous functions
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Absolute continuity on a compact interval
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
Young's inequality for conjugate real exponents
LemmaLibrary prerequisite
Locally integrable functions define distributions
LemmaLibrary prerequisite
Flow calculus of a self-adjoint generator: commutation, invariance and differentiability
LemmaLibrary prerequisite
Time translations and mollification in Bochner Lp
TheoremLibrary prerequisite
Weak unique continuation for bounded potentials in three dimensions
TheoremLibrary prerequisite
Bochner Lp is complete for a Banach value space
DefinitionDependency level 0
Distributional time derivatives with values in a Banach space
DefinitionDependency level 0
Reachability and observation vocabulary for controlled evolutions
DefinitionDependency level 0
Distributional equation convention for complex Schrödinger solutions
DefinitionDependency level 0
Complex integer-order Sobolev notation
DefinitionDependency level 0
Continuous and mixed time-space function spaces
DefinitionDependency level 0
Weak convergence notation in normed state spaces
DefinitionDependency level 0
Periodic energy space on an interval
LemmaDependency level 0
Spectral cutoff cores and unitary identifications of completed scales
LemmaDependency level 0
Compatible Banach pairs and their integrals
LemmaDependency level 0
Continuous free convolution on a finite time interval
LemmaDependency level 0
Weak shifted Dirichlet form and energy bound on an open set
LemmaDependency level 0
Complex polynomial weak derivatives and zero boundary preservation
LemmaDependency level 0
Uniform localized H2 algebra on a smooth exterior domain
LemmaDependency level 0
Helmholtz continuation through a harmonic lift
LemmaDependency level 0
Observation lower bounds give Hilbert control right inverses
LemmaDependency level 0
Realification of Hilbert pairings and conjugate-linear coefficients
PropositionDependency level 0
Independent generalized reflected-ray and GCC convention
PropositionDependency level 0
Star-shaped multiplier signs
DefinitionDependency level 1
Exterior and truncated obstacle domains
LemmaDependency level 1
Compact normalized compressed phase sets over bounded regions
LemmaDependency level 1
An outward Euclidean exit cannot return
LemmaDependency level 1