Alphabeta MathDual Spaces Adjoint Operators and Annihilators — Examples
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Compactness
- 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 Adjoint Operators and Annihilators
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Dimensional Normed Spaces and Riesz Lemma
- Foundations of the Real Numbers for Analysis
- Geometric Hahn Banach and Convex Separation
- Hausdorff via the Diagonal
- 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
- Order, Zorn's Lemma, and the Axiom of Choice
- Outer Measure and the Caratheodory Extension Theorem
- Partitions of Unity and Paracompactness
- 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
- Radon Measures and the Riesz Markov Kakutani Theorem
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Separation Axioms: the Hierarchy
- 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 Cantor Set, Baire Category, and Measure Zero in ℝ
- 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
- Urysohn's Lemma and the Tietze Extension Theorem
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Remark
Two different Riesz representation theorems
Example
Coordinate functionals on sequence spaces
Example
Transposes of the right and left shifts
Example
Evaluation functionals and point masses
Example
The annihilator of a coordinate subspace
Example
Finite-dimensional duals and matrix transposes
Counterexample
The dual construction reverses arrows
Counterexample
The canonical bidual map of c0 misses the constant sequence
Counterexample
The transpose of an injective map need not have norm-dense range
Counterexample
Injective transpose does not imply surjectivity
Sources
- Bühler–Salamon, Functional Analysis, Examples 1.32 and 1.37, pp.32,37
- Bühler–Salamon, Functional Analysis, Examples 1.35–1.36, pp.36–37
- Bühler–Salamon, Functional Analysis, Definition 4.1 and Examples 1.35–1.36, pp.172,36–37
- Bühler–Salamon, Functional Analysis, Example 1.37 and Example 4.4, pp.37,173
- Bühler–Salamon, Functional Analysis, Example 1.36, pp.36–37; Brezis §1.3 notation p.9
- Bühler–Salamon, Functional Analysis, Example 4.5, p.173
- Bühler–Salamon, Functional Analysis, Definition 4.1 and Lemma 4.3(i), pp.172–173
- Bühler–Salamon, Functional Analysis, Examples 1.35–1.36 and §2.4.1, pp.36–37,88
- Bühler–Salamon, Functional Analysis, Example 4.10, p.174
- Bühler–Salamon, Functional Analysis, Example 4.9, p.174; Brezis Remark 20, p.48