Alphabeta Math

Approximation and Compactness in C(K): Examples and Counterexamples

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1 · Prerequisites
Abstract Algebra
Combinatorics
Linear Algebra
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Counterexample

Rudin's bounded rational spikes are not equicontinuous and have no uniformly convergent subsequence

Counterexample

All constant functions form an equicontinuous family that is not pointwise bounded

Counterexample

Translations of a fixed bump on ℝ are uniformly bounded and equicontinuous but have no uniformly convergent subsequence

Example

The Bernstein polynomials of f(x)=x² equal x²+x(1-x)/n

Example

Under the Axiom of Countable Choice and the Axiom of Dependent Choice, the family x↦|x-a|, a∈[0,1], is compact in C([0,1])

Counterexample

Even polynomials on [-1,1] form a unital algebra but are not dense because they do not separate -1 and 1

Counterexample

Polynomials vanishing at zero separate points of [0,1] but are not dense in C([0,1])

Sources