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A unital point-separating real subalgebra of
Definition
Let be a nonempty compact metric space. A set is a unital real function algebra when it contains every constant function and is closed under pointwise addition, scalar multiplication, and multiplication. It separates points when for every distinct there is with . The ambient function space is The space of continuous real-valued functions on a nonempty compact metric space.
Depends on
Used by
- 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]) Counterexample
- A unital separating real function algebra interpolates arbitrary values at two distinct points Lemma
- The uniform closure of a unital real function algebra is closed under absolute value, maximum, and minimum Lemma
- The general real function-algebra definition agrees with the published compact-metric definition Proposition
Dependency tree · two levels
5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- The Stone--Weierstrass Theorem and its Applications (standard reference, not scraped)