Alphabeta MathArticle items · 06 — Square-summable orthogonal families have norm-convergent finite sums
Presents 29 items from the article and its immediate prerequisites, covering published library prerequisites and dependency levels 0–1.
1 · Prerequisites
- Square-summable families on an arbitrary index set and the space ℓ²(I)
- Pythagoras and finite orthogonal sums
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- Hilbert space
- The Axiom of Countable Choice (AC_ω)
- For every ε > 0 in a complete ordered field there is a natural n ≥ 1 with 1/n < ε
- Squaring is monotone on the nonnegatives
- The reverse triangle inequality in a normed space
- Orthonormal families, complete orthonormal systems and Hilbert bases
- The Axiom of Choice
- Adjoint of a densely defined operator
- Symmetric, self-adjoint and essentially self-adjoint operators
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- Projection valued measure
- The Borel sigma-algebra of a topological space
- Probability measures and probability spaces
- Scalar and complex measures from a pvm
- Density operators and scalar spectral probability
- Rank-one operator calculus
- Trace is absolutely convergent and basis independent
- Trace class is a two sided Banach operator ideal
- Non-negative and positive operators
- State and composition postulate P1: Hilbert spaces, preparations and composite systems
- Dimensions, SI unit conventions and quantity types of the framework
- Riesz representation for Hilbert spaces
- Fundamental theorem of calculus for Banach-valued continuous curves
- Linearity of the Bochner integral
- A bounded linear operator between normed spaces
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Bochner integral norm inequality
- Bochner integrability criterion
- Finite spin matrices
- Finite spin-block decomposition
- The elementary tensors of two bases form the product basis of the tensor product
- Real and complex inner product spaces, with the inner product linear in the first argument
- Linear map between vector spaces over the same field
- Linear subspace of a vector space
- The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V
- Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space
- Orthogonality and the orthogonal complement
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- The natural numbers ℕ (von Neumann)
- A finite sum in a commutative monoid indexed by an arbitrary finite set
- The factorial n! and the falling factorial n^k̲, defined by recursion in ℕ
- The induced length is a norm
- Hilbert tensor construction: inner product, completion and product bases
- Transverse one-photon space and helicity projectors
- A smooth bump between concentric Euclidean balls
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- 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
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- AC implies DC implies countable choice
- Stone's theorem: unitary groups and self-adjoint generators
- Stone's theorem: unitary groups and self-adjoint generators
- The Axiom of Choice
- The Axiom of Countable Choice (AC_ω)
- Hilbert space
- A bounded linear operator between normed spaces
- The Hilbert-space adjoint of a bounded operator
- Symmetric, self-adjoint and essentially self-adjoint operators
- Unbounded linear operators: domain, graph and extension
- Schwartz/distribution spaces and Hilbert Fock conventions
- Greatest lower bound (infimum)
- Every nonempty set bounded below has an infimum
- Bosonic and fermionic Fock operators with exact domains
- One-mode factorization of symmetric Fock space
- Hermite recurrence
- Hermite completeness
- The p-series for a real exponent p converges exactly when p is greater than one
- Gottesman, Kitaev and Preskill, Encoding a qubit in an oscillator · Section IV, Eq. (27), position-space codewords with code dimension n=2; Section V, Eqs. (30)–(31) and the paragraph following Eq. (34).
- Complementary channels and coherent information
- Partial trace over the first tensor factor
- Finite Kraus representation
- Finite-dimensional quantum channel
- Quantum code as an isometric encoding and its code projector
- Density operator
- Ordered tensor registers, their inner product, and wire permutations
- Partial trace over a finite tensor factor
- Completely positive maps
- Finite instruments and conditional states
- Finite GKSL generators and units
- Probability measures and probability spaces
- Random elements and real random variables
- The Borel sigma-algebra of a topological space
- Nonempty intersections of sigma-algebras are sigma-algebras, so the generated sigma-algebra exists and is minimal
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Continuous quantum error correction via quantum feedback control · Section II A–B, PDF pp. 2–3, equations (1)–(14).
- Tensor products of operator matrix units form a basis
- A simple formula for the average gate fidelity of a quantum dynamical operation · p. 1, paragraph defining entanglement fidelity; p. 2, paragraph preceding Eq. (11)
- Finite completely positive trace-preserving maps
- Finite GKSL generates CPTP semigroups
- Every finite-dimensional normed space is Banach
- The exponential series of a bounded operator
- Real and complex inner product spaces, with the inner product linear in the first argument
- The standard formulas ⟨ x,y⟩=∑_k<nxₖ yₖ on ℝⁿ and ∑_k<nxₖyₖ on ℂⁿ are inner products
- The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- Linear map between vector spaces over the same field
- Coordinate columns [v]_B and matrices [T]_B^C of linear maps relative to ordered bases
- Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes
- The transpose A^T of a matrix
- Choi matrix with input factor first
- Tensor products of finitely many linear maps
- A finite list v : n → V is an ordered basis if and only if every x ∈ V equals ∑_i<n λᵢ vᵢ for exactly one λ : n → F; those scalars are the coordinates of x in that ordered basis
- Banach space
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- A chosen algebraic basis identifies a finite-dimensional normed space with a coordinate space
- Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces
- Every linear map between finite-dimensional inner product spaces has a unique adjoint
- Theory of Quantum Error Correction for General Noise · PDF p. 3, Noiseless Subsystems
- Exact recovery of an encoded quantum code
- An isometric quantum encoding gives an orthogonal code projector
- For A∈ M_m× n(F) and B∈ M_n× m(F), tr(AB)=tr(BA)
- Non-negative and positive operators
- The direct sum of an indexed family of modules
- Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely
- Fourier inversion on Schwartz space
- Schwartz space and its seminorms
- Integrable real and complex functions, and their integrals
- Every at most countable subset of ℝⁿ is Lebesgue null; in particular λ₁(ℚ)=0
- A nonnegative integral over a null set vanishes
- Universal Lindblad equation for open quantum systems · p. 9, Eqs. (25)–(26) and surrounding text
- Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space
- The orthogonal complement W^⊥={v:⟨ v,w⟩=0 for all w∈ W}
- The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥
- Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases
- Eigenvalues, eigenvectors, eigenspaces E_λ(T)=ker(T-λ I), and the spectrum σ_F(T) of an endomorphism
- Kernel and image of a linear map
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Linear subspace of a vector space
- Linear combination of a finite list, and the span span(S) as the smallest linear subspace containing S
- Vector space over a field
- Balanced maps from a right module and a left module, and bilinear maps over a commutative ring
- Nonnegative real square roots exist uniquely
- Dedekind-complete ordered real field
- The Cauchy-sequence reals have the least-upper-bound property
- 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)
- Dominated convergence
- The modulus of an integral is bounded by the integral of the modulus
- The Lebesgue integral is linear on L¹(μ)
- Integral over a measurable subset
- Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans
- Orthogonal projection is linear, and an orthonormal basis (eᵢ) of W gives P_Wv=∑ᵢ⟨ v,eᵢ⟩ eᵢ
- The real exponential function and the number e by a power series
- The exponential function is strictly increasing
- The complex exponential is entire and its complex derivative is itself
- exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0
- Finite and countable subadditivity of measures
- A measurable function between measurable spaces
- The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- 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, every Borel subset of ℝⁿ is Lebesgue measurable
- Measure-null sets and almost-everywhere statements relative to a measure
- Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space
- The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ
- Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation
- A continuous map has Borel preimages of Borel sets
- Composition with a Borel measurable outer map preserves measurability
- Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined
- 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
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- A map into ℝⁿ is measurable exactly when its coordinates are measurable
- Metric continuity characterisations, with countable choice for the sequential converse
- 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
- Measures on sigma-algebras
- The nonnegative Lebesgue integral
- Real and imaginary parts, complex conjugation, and modulus
- Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
- For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and f(Ā) ⊆ f(A)
- A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice
- Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition
- Trace Norm Rank One Formula
- Trace Norm Duality And Norm Properties
- Trace Norm
- The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
- Inner products separate vectors, and the induced norm is homogeneous: ‖λ v‖=|λ|‖ v‖
- Real and complex inner-product spaces and their induced length
- Adjoints satisfy (S+T)^*=S^*+T^*, (λ T)^*=λ̄ T^*, (ST)^*=T^*S^*, and T^**=T
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- Tensor maps preserve composition, identities, and adjoints
- Finite complex linear isometries preserve inner products
- Dirac Notation Conversion
- Self-adjoint trace-Gibbs normalization and form-energy optimization
- Trace class operator
- Trace of a trace class operator
- Nuclear series characterizes trace norm
- For |r| < 1, ∑_k ≥ 0 rᵏ = 1/(1-r), and for |r| ≥ 1 the series diverges
- Regular Borel measure on an LCH space
- Bounded borel pvm integral
- Integral of a measurable function against a projection-valued measure
- Schwarz and trace-norm channel contraction
- Finite matrix trace-norm ideal inequalities
- Finite registers and their ordered computational bases
- A finite Kraus sum is completely positive
- Composition satisfies ‖ST‖≤‖S‖ ‖T‖
- The induced length is a norm
- The standard list e : n → Fⁿ with eᵢ(i) = 1_F and eᵢ(j) = 0_F for j ≠ i is an ordered basis of Fⁿ; hence dim_F Fⁿ = n, and F⁰ is the zero space with basis ∅ and dimension 0
- Every finite-dimensional normed space is Banach
- A linear map from a finite-dimensional normed space is bounded
- 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
- Continuity and derivatives of positive-base real powers
- On an interval I, for f continuous on I and differentiable at every interior point: f' ≥ 0 throughout gives f nondecreasing, f' > 0 gives f increasing, f' ≤ 0 and f' < 0 give the two decreasing forms; conversely a nondecreasing f has f' ≥ 0 and a nonincreasing f has f' ≤ 0 wherever it is differentiable, and no strict converse is claimed
- 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
- The ε-δ limit lim_x → c f(x) = L of f : A → ℝ at a limit point c of A
- The extended real line ℝ̄ = ℝ ∪ {-∞, +∞}, its order, and the arithmetic that is left undefined
- The polar surface set function on the unit sphere
- Sphere integrals in angle coordinates
- 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)
- The derivatives of sine and cosine are cosine and minus sine
- The addition formulas for sine and cosine
- Parity and the Pythagorean identity for sine and cosine
- Fubini's theorem for L¹ functions on a sigma-finite product
- A function differentiable at c is continuous at c
- A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
- A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral
- Quarter-turn values and shifts by pi/2 and pi
- 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 box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Two integrable functions are equal almost everywhere exactly when all of their indefinite integrals agree
- The nonnegative integral agrees with the simple integral on simple functions
- The integral of a nonnegative simple function
- On Borel subsets of ℝᵐ⁺ⁿ, the product λₘ × λₙ agrees with λₘ₊ₙ
- The Borel product of ℝᵐ and ℝⁿ is the Borel sigma-algebra of ℝᵐ⁺ⁿ
- The radiation sector of the framework: modes and the number, energy and momentum assignments
- Cutoff dipole relative bound
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Single-mode thermal trace
- Diagonal adjoint and graph-core lemma
- Kato-Rellich theorem
- Transverse continuum free Fock Hamiltonian
- Quantum and Atom Optics · Section 5.1, discussion of the long-wavelength approximation and equation (5.5)
- Quantum Optics · Table 1.2, entry (3), thermal state, PDF page 39
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
Square-summable orthogonal families have norm-convergent finite sums
PostulateLibrary prerequisite
Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes
TheoremLibrary prerequisite
Norm Dyson series and uniqueness
TheoremLibrary prerequisite
Tensor spin composition
TheoremLibrary prerequisite
Transverse rotations preserve helicity and generate total angular momentum
DefinitionDependency level 0
Time-local interaction-picture convention
DefinitionDependency level 0
Bosonic loss, Kerr shifts and finite-code comparison scope
DefinitionDependency level 0
Complementary error records in a Kraus basis
DefinitionDependency level 0
Continuous monitoring versus pulsed syndrome protocols
DefinitionDependency level 0
Entanglement fidelity on a finite logical space
DefinitionDependency level 0
Finite GKSL evolution and the QEC channel interface
DefinitionDependency level 0
Finite operator and ket conventions
DefinitionDependency level 0
Liouville matrix and Choi reshuffling conventions
DefinitionDependency level 0
Passive subspace and subsystem protection interface
DefinitionDependency level 0
Code restriction and decoded channel types
DefinitionDependency level 0
Finite spin blocks, spherical tensors and Clebsch–Gordan phases
DefinitionDependency level 0
Stationary correlation spectra and ULE kernel regularity
DefinitionDependency level 0
Finite support square roots and generalized inverses
LemmaDependency level 0
A scalar normalization survives a two-point functional
LemmaDependency level 0
Finite-cell sums converge on a compact frequency shell
LemmaDependency level 0
An absolute thermal envelope controls radial correlations
LemmaDependency level 0
Reference-uniform trace-norm sandwich estimates
LemmaDependency level 0
Finite-mode bosonic Gibbs occupations and moments
LemmaDependency level 0
Telescoping error budget for repeated finite channels
LemmaDependency level 0
Finite exponential and smooth quotient remainder control
LemmaDependency level 0
Local inversion of a fidelity first crossing
LemmaDependency level 0
Transverse projector and isotropic angular contraction
PostulateDependency level 0
Regularized transverse photon dipole bath model
DefinitionDependency level 1