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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
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 26 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- The Stone--Weierstrass Theorem and its Applications (standard reference, not scraped)