Alphabeta MathDual Spaces Adjoint Operators and Annihilators
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Norms and Realification
- Complex Subspace Realification
- Complexification, Realification and Real Structures
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Geometric Hahn Banach and Convex Separation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Order, Zorn's Lemma, and the Axiom of Choice
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sigma Algebras and Borel Sets
- Signed and Complex Measures Hahn and Jordan
- Simple Field Extensions and the Construction of the Complex Numbers
- Subspaces, Products, and Quotients
- Suprema and Infima
- The Analytic Hahn Banach Theorem
- The Baire Principles of Functional Analysis
- The Derivative and the Mean Value Theorems
- The Duality of Lᵖ and L^q
- The Exponential Function
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Radon Nikodym Theorem and Lebesgue Decomposition
- The Riemann Integral: Definition and Integrability
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Remark
The continuous dual, its completeness, and evaluation
Definition
Annihilator notation and the preannihilator
Lemma
Annihilators and preannihilators are norm closed
Theorem
The dual of a quotient is its annihilator
Theorem
The dual of a closed subspace is a dual quotient
Corollary
Distance to an annihilator is the restriction norm
Definition
The transpose of a bounded operator
Lemma
The transpose is bounded with the same norm
Lemma
Transposition reverses composition
Lemma
Elementary kernel and range annihilator identities
Corollary
Dense range is equivalent to injectivity of the transpose
Definition
The canonical evaluation map into the bidual
Theorem
The canonical bidual map is an isometry
Corollary
Distance to a closed subspace via unit annihilators
Lemma
The canonical map is natural
Definition
Reflexivity is surjectivity of the canonical map
Lemma
Membership in the transpose range by an operator estimate
Lemma
Closed range is equivalent to a quotient estimate
Lemma
A lower bound for the transpose forces a dense image of a ball
Theorem
Surjectivity is equivalent to a lower bound for the transpose
Theorem
Banach closed-range theorem
Theorem
Bounded below is equivalent to surjectivity of the transpose
Corollary
Invertibility and the inverse of the transpose
Lemma
Finite truncations approximate null and summable sequences
Theorem
The continuous dual of c0 is ell-one
Theorem
The complex continuous dual of ell-one is ell-infinity
Remark
The published Lp duality theorem in abstract notation
Definition
The weak-star topology from finite evaluations
Lemma
Finite evaluations separate a functional from a dual subspace
Theorem
Double annihilators give norm and weak-star closures
Theorem
Kernel-range identities and the weak-star closure of the transpose range
Sources
- Bühler–Salamon, Functional Analysis, §§1.3.1–1.3.2, Theorem 1.31 and (1.22), pp.31–32
- Brezis, Functional Analysis, Sobolev Spaces and PDEs, §1.3, notation p.9
- Brezis, Functional Analysis, Sobolev Spaces and PDEs, §1.3, notation before Proposition 1.9, p.9
- Bühler–Salamon, Functional Analysis, Corollary 2.57(ii), pp.84–85
- Bühler–Salamon, Functional Analysis, Corollary 2.57(i), pp.84–85
- Bühler–Salamon, Functional Analysis, Corollary 2.58, (2.33), p.85
- Bühler–Salamon, Functional Analysis, §4.1.1, Definition 4.1, p.172
- Bühler–Salamon, Functional Analysis, Lemma 4.2, p.172
- Bühler–Salamon, Functional Analysis, Lemma 4.3(i), p.173
- Bühler–Salamon, Functional Analysis, Theorem 4.8(i), pp.174; Corollary 2.55, p.84
- Bühler–Salamon, Functional Analysis, Theorem 4.8(ii), p.174
- Bühler–Salamon, Functional Analysis, §2.4.1, (2.39), p.88
- Bühler–Salamon, Functional Analysis, Lemma 2.68, p.88
- Bühler–Salamon, Functional Analysis, Corollary 2.69, p.88
- Bühler–Salamon, Functional Analysis, Lemma 4.3(ii), p.173
- Bühler–Salamon, Functional Analysis, Definition 2.70, p.89
- Bühler–Salamon, Functional Analysis, Lemma 4.15, pp.175–176
- Bühler–Salamon, Functional Analysis, Theorem 4.16(ii)–(iii), pp.178–179
- Bühler–Salamon, Functional Analysis, Theorem 4.16 proof (vii) => (i), (4.10)–(4.11), p.180
- Bühler–Salamon, Functional Analysis, Corollary 4.17(i), p.181, with Theorem 4.16 p.180
- Bühler–Salamon, Functional Analysis, Theorem 4.16, pp.178–181
- Bühler–Salamon, Functional Analysis, Corollary 4.17(ii), p.181
- Bühler–Salamon, Functional Analysis, Corollary 4.18, p.182
- Bühler–Salamon, Functional Analysis, Examples 1.35–1.36, pp.36–37
- Bühler–Salamon, Functional Analysis, Example 1.36, pp.36–37
- Bühler–Salamon, Functional Analysis, Example 1.35, p.36 (scalar-field extension of its coefficient proof)
- Bühler–Salamon, Functional Analysis, Example 1.33, p.33 and Example 1.35, p.36
- Bühler–Salamon, Functional Analysis, Example 3.9(i), p.122; Lemma 3.6(i), pp.119–120 (real source; complex disk version via the same initial-topology construction)
- Bühler–Salamon, Functional Analysis, Theorem 3.12(ii), pp.125–127, and Corollary 3.26(i), p.130; finite-coordinate adaptation
- Bühler–Salamon, Functional Analysis, Corollary 3.26(i)–(iii), p.130; Corollary 2.55, p.84 (real source; K-linear finite-coordinate proof covers C)
- Bühler–Salamon, Functional Analysis, Theorem 4.8 and Corollary 3.26, pp.174 and 130