Alphabeta MathArticle items · 03 — Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space
Presents 29 items from the article and its immediate prerequisites, covering published library prerequisites.
1 · Prerequisites
- The adjoint T^*:W→ V is characterised by ⟨ Tv,w⟩_W=⟨ v,T^*w⟩_V
- Hilbert-adjoint identities
- The Hilbert-space adjoint of a bounded operator
- The spaces B(X,Y) and B(X) of bounded linear operators
- Real and complex inner-product spaces and their induced length
- The Axiom of Countable Choice (AC_ω)
- Adjoint of a densely defined operator
- Closability is equivalent to density of the adjoint domain
- The adjoint is well defined, closed, and reverses inclusions
- Closure of a closable operator
- Densely defined, closed and closable operators, and cores
- The zero sets of sine and cosine and the least positive common period 2 pi
- A signed measure is countably additive and takes at most one infinite value
- A complex measure is a finite-valued countably additive set function
- The extended real line ℝ̄ = ℝ ∪ {-∞, +∞}, its order, and the arithmetic that is left undefined
- Absolute value and singular values of a compact operator
- Singular value decomposition for compact operators
- Bounded finite rank operators are compact
- If a series converges then its terms tend to 0
- Series and absolute convergence in a normed space
- Compact linear operator
- The operator norm as the least bound and as the unit-sphere or unit-ball supremum
- Convergence of a sequence in a metric space: xₖ → x iff d(xₖ, x) → 0 in ℝ
- Greatest lower bound (infimum)
- Finite-dimensional vector space, and its dimension dim_F V; infinite-dimensional means having no finite basis
- Hilbert space
- The vector space M_m × n(F) := F^ m × n of m by n matrices over a field, with entrywise operations
- A finite sum in a commutative monoid indexed by an arbitrary finite set
- Nuclear series characterizes trace norm
- Square-summable families on an arbitrary index set and the space ℓ²(I)
- The finite Bessel inequality and best approximation by a finite orthonormal family
- Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs
- Orthonormal families, complete orthonormal systems and Hilbert bases
- A bounded linear operator between normed spaces
- Coordinate columns [v]_B and matrices [T]_B^C of linear maps relative to ordered bases
- Similar matrices have the same trace
- Similarity is an equivalence relation, and two matrices represent the same endomorphism in two bases exactly when they are similar
- Linear map between vector spaces over the same field
- Linear subspace of a vector space
- Complete metric space: every Cauchy sequence converges in the space
- Vector-valued functions f : A → ℝᵐ, their limits and continuity, with the dictionary to the metric notions
- A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions
- For n ≥ 1 all norms on ℝⁿ are equivalent
- Equivalent norms, and the dictionary with equivalent metrics
- The derivative f'(c) = lim_x → c f(x) - f(c)/x - c of f : A → ℝ at a point c ∈ A that is a limit point of A, and differentiability on a set
- The lower and upper Darboux integrals of a bounded f on [a,b] as sup_P L(f,P) and inf_P U(f,P), Darboux integrability as their equality, and the notation ∫ₐᵇ f
- The integral with oriented limits: ∫ₐᵃ f := 0 and ∫_bᵃ f := -∫ₐᵇ f
- Integrable functions on [a,b] form a set closed under sums and scalar multiples, and ∫ₐᵇ(λ f+μ g) = λ∫ₐᵇ f + μ∫ₐᵇ g
- 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 sequence in a metric space has at most one limit
- The ε-δ limit lim_x → c f(x) = L of f : A → ℝ at a limit point c of A
- At a limit point of the domain a function has at most one limit
- The vector space F^X of all functions X → F with pointwise operations, and Fⁿ as the case X = n = {0, 1, …, n-1}
- 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
- Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length
- A function integrable on [a,b] is integrable on every closed subinterval
- For a<c<b: f is integrable on [a,b] if and only if it is integrable on [a,c] and on [c,b], and then ∫ₐᵇ f = ∫ₐᶜ f + ∫_cᵇ f; with the oriented form for arbitrary a,b,c
- Limit point, isolated point, adherent point, derived set, and dense subset of ℝ
- The p-norms ‖ x‖ₚ for rational p ≥ 1, and ‖ x‖_∞
- The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ
- Lower bound, bounded below, bounded set
- Real and imaginary parts, complex conjugation, and modulus
- ℂ=ℝ[x]/(x²+1) is a field, every element is uniquely a+bi, and every nonzero element has inverse (a-bi)/(a²+b²)
- The Cauchy-sequence reals have the least-upper-bound property
- Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0}
- Squaring is monotone on the nonnegatives
- Squares of nonzero elements are positive
- A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral
- If u,v are differentiable on [a,b] with u',v' integrable, then ∫ₐᵇ u v' = u(b)v(b)-u(a)v(a) - ∫ₐᵇ u'v
- 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)
- Dominated convergence
- Complex Lp classes and Euclidean test-function conventions
- Temporal point-process models and conditional intensity
- Absolute continuity on a compact interval
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- The indefinite integral of an L¹ function is absolutely continuous
- The indefinite integral of an L¹ function is differentiable almost everywhere
- 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)
- The exponential function is smooth and (exp)'=exp
- An absolutely continuous function with zero derivative almost everywhere is constant
- The product of two absolutely continuous functions is absolutely continuous
- The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)
- Continuity from above when one set has finite measure
- The Axiom of Choice
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Dedekind-complete ordered real field
- Continuity and derivative on a real interval
- Dedekind completeness implies the Archimedean property
- Closed real intervals are compact by completeness
- Continuous functions on a closed interval attain their extrema
- Real and complex modulus product and triangle inequalities
- The operator norm is a norm on the space of bounded linear operators
- Composition satisfies ‖ST‖≤‖S‖ ‖T‖
- If Y is Banach then B(X,Y) is Banach
- Series criterion for Banach spaces
- An absolutely convergent series has Cauchy partial sums
- The real exponential function and the number e by a power series
- The exponential is a continuous bijection from ℝ onto (0,∞)
- C star algebra
- The induced length is a norm
- Every finite-dimensional normed space is Banach
- The tensor product M⊗_R N from the additive group underlying the free ℤ-module on M× N, elementary tensors, and finite tensor sums
- The norm completion of an inner-product space is a Hilbert space
- Completion of a normed space
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Every finite-dimensional real or complex inner product space has an orthonormal basis
- Parseval equivalences for an orthonormal family
- Orthogonality and the orthogonal complement
- Schwartz/distribution spaces and Hilbert Fock conventions
- Bounded tensor maps: extension, norm estimate, adjoint and commutation
- Permutation sector projections on the n-fold tensor power
- Schwartz space and its seminorms
- Tonelli's theorem for nonnegative measurable functions on a sigma-finite product
- On Borel subsets of ℝᵐ⁺ⁿ, the product λₘ × λₙ agrees with λₘ₊ₙ
- Unbounded Borel functional calculus: domains, products, spectral mapping
- Spectral theorem for unbounded self-adjoint operators (PVM form)
- Integral of a measurable function against a projection-valued measure
- Infinitesimal generator of a unitary group
- Fatou's lemma
- The unbounded PVM integral is densely defined, closed and normal
- Inner products separate vectors, and the induced norm is homogeneous: ‖λ v‖=|λ|‖ v‖
- The real Frechet derivative on real and complex matrix spaces with the Frobenius norm
- Rectangular matrix multiplication and the identity matrix Iₙ, including zero-sized shapes
- In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix
- The Frobenius norm ‖ A‖_F=(∑_i,j|aᵢⱼ|²)^1/2 on real or complex matrices
- The orthogonal projection P_Wv is the W-component in V=W⊕ W^⊥
- Bessel's inequality for a finite orthonormal list and Parseval's identity for an orthonormal basis
- Real and complex inner product spaces, with the inner product linear in the first argument
- Finite-dimensional Riesz representation: every functional is uniquely v↦⟨ v,w⟩
- 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
- Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero
- If lim_x → c f(x) = L ≠ 0 then |f| > |L|/2 on a punctured neighbourhood of c; in particular if L > 0 then f > L/2 > 0 there
- The ε-neighbourhood and the punctured ε-neighbourhood of a point of ℝ
- Basic properties of the absolute value
- The reverse triangle inequality
- Maximum and minimum of a set
- Every nonempty finite set of reals has a maximum and a minimum
- Integer powers aᵐ
- Ordered field
- Field
- The complex exponential by its power series
- The complex exponential series converges absolutely for every complex argument
- The Cauchy product of two absolutely convergent complex series converges absolutely to the product of their sums
- The binomial theorem over the complex field
- C(n, k) k! (n-k)! = n! for k ≤ n; hence C(n, k) k! = n^k̲, the quotient n!/(k!(n-k)!) is a natural number, and C(n, k) = C(n, n-k)
- Continuous first partial derivatives and the Cauchy–Riemann equations imply complex differentiability pointwise, and holomorphy when they hold throughout an open set
- exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0
- The derivatives of sine and cosine are cosine and minus sine
- A function differentiable at c is continuous at c
- Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined
- Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with ∂_z̄f=0, or with the Cauchy–Riemann equations
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
Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space
DefinitionLibrary prerequisite
Self-adjoint, positive, unitary and normal operators
DefinitionLibrary prerequisite
Symmetric, self-adjoint and essentially self-adjoint operators
DefinitionLibrary prerequisite
Tangent, cotangent, secant, and cosecant on their exact natural domains
DefinitionLibrary prerequisite
The total variation |nu|(E) from countable measurable partitions
DefinitionLibrary prerequisite
Trace class operator
DefinitionLibrary prerequisite
The trace tr(A) as the sum of the diagonal entries
DefinitionLibrary prerequisite
Trace of a trace class operator
DefinitionLibrary prerequisite
The basis-independent trace of an endomorphism of a finite-dimensional vector space
DefinitionLibrary prerequisite
Unbounded linear operators: domain, graph and extension
DefinitionLibrary prerequisite
The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral
LemmaLibrary prerequisite
Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive
LemmaLibrary prerequisite
Complex integration by parts on intervals and decaying lines
LemmaLibrary prerequisite
Inter-event survival and density from a conditional intensity
LemmaLibrary prerequisite
Continuous functions have an oriented Riemann integral
LemmaLibrary prerequisite
The exponential series of a bounded operator
LemmaLibrary prerequisite
Finite matrix algebras with the operator norm are C star algebras
LemmaLibrary prerequisite
Hilbert tensor construction: inner product, completion and product bases
LemmaLibrary prerequisite
Bosonic and fermionic Fock operators with exact domains
LemmaLibrary prerequisite
The oriented Riemann integral is linear in continuous integrands
LemmaLibrary prerequisite
Schwartz derivatives are integrable
LemmaLibrary prerequisite
A self-adjoint operator generates a strongly continuous unitary group
PropositionLibrary prerequisite
Adjoints satisfy (S+T)^*=S^*+T^*, (λ T)^*=λ̄ T^*, (ST)^*=T^*S^*, and T^**=T
PropositionLibrary prerequisite
Matrix differentials obey the sum rule, product rule, and adjoint rule
PropositionLibrary prerequisite
Orthogonal projection is linear, and an orthonormal basis (eᵢ) of W gives P_Wv=∑ᵢ⟨ v,eᵢ⟩ eᵢ
TheoremLibrary prerequisite
Every linear map between finite-dimensional inner product spaces has a unique adjoint
TheoremLibrary prerequisite
Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function
TheoremLibrary prerequisite
exp(z+w)=exp z exp w, and the complex exponential extends the real exponential
TheoremLibrary prerequisite