Alphabeta MathArticle items · 02 — Greatest lower bound (infimum)
Presents 28 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- Lower bound, bounded below, bounded set
- Sesquilinear and Hermitian forms over a field with an involution, using the convention linear in the first variable
- Real and imaginary parts, complex conjugation, and modulus
- Closure properties of measurable functions used by the integral
- The nonnegative Lebesgue integral
- Lebesgue outer measure on ℝⁿ
- Carathéodory measurable sets
- The norm ‖ v‖=√⟨ v,v⟩ induced by a real or complex inner product
- Linear map between vector spaces over the same field
- The exponential is a continuous bijection from ℝ onto (0,∞)
- A bounded linear operator between normed spaces
- The Cauchy-sequence reals have the least-upper-bound property
- Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis
- The Borel product of ℝᵐ and ℝⁿ is the Borel sigma-algebra of ℝᵐ⁺ⁿ
- The Borel sigma-algebra of a subspace is the trace of the ambient Borel sigma-algebra
- Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable
- A continuous map has Borel preimages of Borel sets
- The Borel sigma-algebra of a topological space
- The Axiom of Countable Choice (AC_ω)
- Tangent is a continuous strictly increasing bijection from (-π/2,π/2) onto ℝ
- Continuous inverse theorem: a continuous injective f on an interval I is a bijection onto the order-convex set f[I], and the inverse g : f[I] → I is continuous and strictly monotone in the same sense as f
- Measures on sigma-algebras
- Measure spaces
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Finite registers and their ordered computational bases
- Ordered tensor registers, their inner product, and wire permutations
- Tensor products of finitely many linear maps
- Non-negative and positive operators
- The basis-independent trace of an endomorphism of a finite-dimensional vector space
- Kernel and image of a linear map
- Linear subspace of a vector space
- 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
- The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V
- Agreement on pure states implies complete agreement
- Every linear map between finite-dimensional inner product spaces has a unique adjoint
- An isometric quantum encoding gives an orthogonal code projector
- Trace preservation is Kraus normalization
- Finite iterated tensor products represent multilinear maps independently of parenthesization
- Every finite-dimensional real or complex inner product space has an orthonormal basis
- Partial trace over a finite tensor factor
- The partial trace is characterized by the trace pairing and is basis-independent
- Tensor maps preserve composition, identities, and adjoints
- Finite wire permutations are unitary basis permutations
- Completely positive maps
- Finite positive effect family
- A complete positive family consists of effects
- The space L(V,W) of linear maps with pointwise addition and scalar multiplication
- L(V,W) is a vector space over the common scalar field
- Identity maps and composites of linear maps are linear
- The singular values of a linear map as the eigenvalues of the positive square root of T^*T
- Singular values are well defined because the positive square root of T^*T is unique
- Schwartz space and its seminorms
- Test function space d of an open set
- Test function topology
- Tempered distribution
- Hilbert space
- Hilbert tensor construction: inner product, completion and product bases
- Bounded tensor maps: extension, norm estimate, adjoint and commutation
- Permutation sector projections on the n-fold tensor power
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- A measurable function between measurable spaces
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
DefinitionLibrary prerequisite
Greatest lower bound (infimum)
DefinitionLibrary prerequisite
Real and complex inner product spaces, with the inner product linear in the first argument
DefinitionLibrary prerequisite
Integrable real and complex functions, and their integrals
DefinitionLibrary prerequisite
Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
DefinitionLibrary prerequisite
Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces
DefinitionLibrary prerequisite
The natural logarithm as the inverse of the exponential function
DefinitionLibrary prerequisite
The operator norm as the least bound and as the unit-sphere or unit-ball supremum
DefinitionLibrary prerequisite
The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖
DefinitionLibrary prerequisite
Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases
DefinitionLibrary prerequisite
The polar surface set function on the unit sphere
DefinitionLibrary prerequisite
The principal inverse tangent arctan:ℝ→(-π/2,π/2)
DefinitionLibrary prerequisite
Probability measures and probability spaces
DefinitionLibrary prerequisite
Choi matrix with input factor first
DefinitionLibrary prerequisite
Density operator
DefinitionLibrary prerequisite
Dirac Notation Conversion
DefinitionLibrary prerequisite
Exact recovery of an encoded quantum code
DefinitionLibrary prerequisite
Partial trace over the first tensor factor
DefinitionLibrary prerequisite
Finite instruments and conditional states
DefinitionLibrary prerequisite
Finite Kraus representation
DefinitionLibrary prerequisite
Finite POVMs and their outcome law
DefinitionLibrary prerequisite
Finite projective measurements
DefinitionLibrary prerequisite
Pure state and ray
DefinitionLibrary prerequisite
Finite-dimensional quantum channel
DefinitionLibrary prerequisite
Quantum code as an isometric encoding and its code projector
DefinitionLibrary prerequisite
Trace Norm
DefinitionLibrary prerequisite
Schwartz/distribution spaces and Hilbert Fock conventions
DefinitionLibrary prerequisite
Random elements and real random variables
DefinitionLibrary prerequisite