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The space of continuous real-valued functions on a nonempty compact metric space
Definition
Let be a nonempty compact metric space (Open cover, subcover, compact metric space, and compact subset of a metric space). Define
where is the function space of The vector space of all functions with pointwise operations, and as the case and is the usual metric (The absolute value makes a metric space: is a metric, its open balls are the intervals , and it is unbounded, Continuity of a map between metric spaces, at a point and globally, in the - form).
This definition introduces the set of continuous functions only. Boundedness and the supremum metric are assertions to be proved, not clauses of the definition.
Depends on
- The vector space $F^{X}$ of all functions $X \to F$ with pointwise operations, and $F^{n}$ as the case $X = n = \{0, 1, \dots, n-1\}$
- Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form
- Open cover, subcover, compact metric space, and compact subset of a metric space
- The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded
Used by
- A unital point-separating real subalgebra of C(K,ℝ) Definition
- Equicontinuity, pointwise boundedness, and uniform boundedness for families in C(K,ℝ) Definition
- Unique ergodicity Definition
- An equicontinuous pointwise-bounded family in C(K,ℝ) has a finite net in the supremum metric Lemma
- Continuous functions determine Borel probabilities on compact metric spaces Lemma
- Functions satisfying a fixed local Lipschitz bound somewhere form a closed subset of C([0,1]) Lemma
- Polygonal functions with sufficiently steep nonvertex slopes are dense in C([0,1]) Lemma
- The general real function-algebra definition agrees with the published compact-metric definition Proposition
- C(K,ℝ) is complete in the supremum metric for every nonempty compact metric space K Theorem
Dependency tree · two levels
26 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
- MIT OpenCourseWare 18.100B, Real Analysis, Lectures 20–21 (standard reference, not scraped)
- W. Trench, Introduction to Real Analysis (standard reference, not scraped)