Alphabeta MathStone–Weierstrass in General: Examples and Counterexamples
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Approximation and Compactness in C(K)
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- 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
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Function Space Topologies and the Exponential Law
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Stone–Weierstrass in General
- Subspaces, Products, and Quotients
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Counterexample
The disc algebra is unital and separating but not self-adjoint or dense
Example
Trigonometric polynomials are uniformly dense on the unit circle
Example
The lattice generated by the constants and the distance functions is dense on every compact metric space
Example
On a finite compact Hausdorff space a unital separating algebra contains every scalar-valued function
Example
Endpoint-duplicating functions on [0,1] become all continuous functions on the endpoint quotient
Example
The polynomial algebra is dense but not closed on a nondegenerate compact interval
Corollary
A continuous real function on [0,1] whose every moment ∫₀¹ xⁿf vanishes is identically zero
Corollary
Every continuous function on [0,1] is uniformly approximated by everywhere-differentiable functions whose derivative vanishes at a prescribed point
Sources
- E. Carlen, Notes on Topology and the Stone-Weierstrass Theorem, unnumbered counterexample in Section 1.6
- E. Carlen, Notes on Topology and the Stone-Weierstrass Theorem, Theorem 1.30
- E. Carlen, Notes on Topology and the Stone-Weierstrass Theorem, Lemma 1.27
- J. M. Erdman, A Companion to Real Analysis, Theorem 21.2.15
- J. M. Erdman, A Companion to Real Analysis, Example 21.2.1 and Corollary 21.2.7
- J. M. Erdman, A Companion to Real Analysis, Proposition 21.2.9
- J. M. Erdman, A Companion to Real Analysis, Proposition 21.2.10