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Article items · 01 — Cauchy-Schwarz inequality for $L^2$
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Holder's inequality for integrals, including the endpoint cases
- Equality in Holder's inequality for 1 < p < ∞
- Every finite-dimensional normed space is Banach
- Normed subspace
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- ℚ is countably infinite
- ℕ × ℕ ≈ ℕ
- Both ℚ and ℝ ∖ ℚ are dense in ℝ, and every nonempty open subset of ℝ is uncountable
- The recursion theorem
- The well-ordering principle
- For every ε > 0 in a complete ordered field there is a natural n ≥ 1 with 1/n < ε
- ℂ is the real coordinate plane, with coordinate arithmetic
- Complex norm and complex normed space
- A complex normed linear subspace
- Realification of a complex vector space by restriction of scalars
- Realification preserves a complex norm and its metric
- A complex normed subspace and its finite basis realify
- Riesz representation for Hilbert spaces
- The canonical evaluation map into the bidual
- Reflexivity is surjectivity of the canonical map
- The dual space X^* of a normed space and its dual norm
- Real and complex inner-product spaces and their induced length
- The closed unit ball is compact if and only if the normed space is finite-dimensional
- χ_A(x) is monic of degree n; for n≥1 its xⁿ⁻¹ coefficient is -tr(A) and its constant coefficient is (-1)ⁿ det(A), while χ_0×0=1
- For every finite-dimensional space, σ_F(T) is exactly the set of roots in F of χ_T
- The basis-independent characteristic polynomial χ_T of an endomorphism of a finite-dimensional space, including χ_T=1 in dimension zero
- Reflexive iff unit ball weakly compact
- Eberlein–Šmulian theorem
- Weak topology on a normed space
- Relative dual norming, point separation, and recovery of the norm
- The real dominated-extension principle as an additional hypothesis over ZF
- On a convex open set, a uniform bound ‖Df(z)v‖₂≤ M‖v‖₂ implies ‖f(y)-f(x)‖₂≤ M‖y-x‖₂
- Unbounded linear operators: domain, graph and extension
- Densely defined, closed and closable operators, and cores
- Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets
- Hilbert space
- Field
- Evaluation and roots of a polynomial in a commutative target ring
- Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree
- Choice function
- A norm on a real vector space, the induced metric, and the dictionary with the metric axioms
- Complete metric space: every Cauchy sequence converges in the space
- Measure spaces
- Strongly measurable Banach-valued function
- The Banach-valued simple integral is well defined
- Cᵏ maps and multi-index derivative notation in Euclidean space
- The total (Fréchet) derivative Df(a) as the linear first-order approximation with o(‖h‖₂) remainder
- Real and complex inner product spaces, with the inner product linear in the first argument
- Linear map between vector spaces over the same field
- The support of a function on ℝⁿ and its compactly supported Riemann integral
- Vector-valued functions f : A → ℝᵐ, their limits and continuity, with the dictionary to the metric notions
- Specified finite piecewise C1 boundary presentations
- Exponential bound for a C0-semigroup
- Fundamental theorem of calculus for Banach-valued continuous curves
- Infinitesimal generator of a C0-semigroup
- Strongly continuous semigroup
- The variation-of-constants integral is continuous for integrable forcing
- Fréchet derivative between Banach spaces
- Continuity of a map between metric spaces, at a point and globally, in the ε-δ form
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies
- Finite, countably infinite, countable, uncountable
- Open cover, subcover, compact metric space, and compact subset of a metric space
- ℂ=ℝ[x]/(x²+1) is a field, every element is uniquely a+bi, and every nonzero element has inverse (a-bi)/(a²+b²)
- Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0}
- The Cauchy-sequence reals have the least-upper-bound property
- The space Lᵖ(μ) as the quotient by null functions
- The essential supremum of a measurable function with respect to a measure
- The class L¹(μ) of integrable functions
- The space C₀(ℝⁿ) of continuous functions vanishing at infinity
- Arithmetic and lattice operations preserve measurability whenever they are defined
- Threshold characterisations of real-valued and extended-real-valued measurability
- The Dirac set function at a point
- A Dirac set function is a probability measure
- Every at most countable subset of ℝⁿ is Lebesgue null; in particular λ₁(ℚ)=0
- 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
- Integrable real and complex functions, and their integrals
- The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution
- The quotient ring R/I with (r+I)(s+I)=rs+I
- Field extensions, generated subrings F[S], generated subfields F(S), and simple extensions
- Continuity of a map of topological spaces at a point and globally
- 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
- Separated sets, disconnection, and connected subset of ℝ
- The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- A nonempty set is at most countable iff it is a surjective image of ℕ
- 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
- If V has a basis with n elements and a basis with m elements then n = m; and if V has one finite basis then every basis of V is finite
- 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
- Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S
- A field is a vector space over itself, and over any subfield K ⊆ F every F-vector space is a K-vector space by restricting the scalars
- Subfield: a subring of a field closed under inverses of its nonzero elements, and therefore a field with the restricted operations
- Vector space over a field
- Equinumerous sets, A ≈ B and A ⪯ B
- Injection, surjection, bijection
- The pigeonhole principle on ℕ
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
Cauchy-Schwarz inequality for L²
CorollaryLibrary prerequisite
A finite-dimensional normed subspace is closed
CorollaryLibrary prerequisite
Hilbert spaces are reflexive
CorollaryLibrary prerequisite
In an infinite-dimensional normed space the closed unit ball is not compact
CorollaryLibrary prerequisite
Every endomorphism of a nonzero finite-dimensional vector space over an algebraically closed field has an eigenvalue
CorollaryLibrary prerequisite
Reflexivity is equivalent to weak subsequential compactness of bounded sequences
CorollaryLibrary prerequisite
A totally differentiable map with zero derivative on a convex open set is constant
DefinitionLibrary prerequisite
Adjoint of a densely defined operator
DefinitionLibrary prerequisite
An algebraically closed field: every nonconstant polynomial has a root in the field
DefinitionLibrary prerequisite
The Axiom of Choice
DefinitionLibrary prerequisite
Banach space
DefinitionLibrary prerequisite
Banach-valued simple function and integral
DefinitionLibrary prerequisite
Bochner-integrable function
DefinitionLibrary prerequisite
Bounded Cᵏ domains and boundary charts
DefinitionLibrary prerequisite
Bounded C1 domains and their outward normals
DefinitionLibrary prerequisite
A bounded linear operator between normed spaces
DefinitionLibrary prerequisite
The spaces C_c(ℝⁿ) and C_c^∞(ℝⁿ)
DefinitionLibrary prerequisite
Classical normal derivative
DefinitionLibrary prerequisite
Classical, strong and mild abstract Cauchy solutions
DefinitionLibrary prerequisite
Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
DefinitionLibrary prerequisite
Real and imaginary parts, complex conjugation, and modulus
DefinitionLibrary prerequisite
Complex Lp classes and Euclidean test-function conventions
DefinitionLibrary prerequisite
The complex numbers as ℝ[x]/(x²+1), with the real embedding and imaginary unit i
DefinitionLibrary prerequisite
Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
DefinitionLibrary prerequisite
A convex subset of ℝᵐ contains every line segment between two of its points
DefinitionLibrary prerequisite
The Axiom of Countable Choice (AC_ω)
DefinitionLibrary prerequisite
The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
DefinitionLibrary prerequisite