equicontinuous
1 Definition
Let be a topological space, a metric space and the set of continuous functions .
Let be a subset of . A function is continuous at a point when given there is a neighbourhood of such that for every . When the same neighbourhood can be chosen for all functions , the family is said to be equicontinuous. More precisely:
Definition - Let be a subset of . The set of functions is said to be equicontinuous at if for every there is a neighbourhood of such that for every and every we have
The set is said to be equicontinuous if it is equicontinuous at every point .
2 Examples
-
•
A finite set of functions in is always equicontinuous.
-
•
When is also a metric space, a family of functions in that share the same Lipschitz constant is equicontinuous.
-
•
The family of functions , where is given by is not equicontinuous at .
3 Properties
-
•
If a subset is totally bounded under the uniform metric, then is equicontinuous.
-
•
Suppose is compact. If a sequence of functions in is equibounded and equicontinuous, then the sequence has a uniformly convergent subsequence. (ArzelÃÂ ’s theorem (http://planetmath.org/AscoliArzelaTheorem))
-
•
Let be a sequence of functions in . If is equicontinuous and converges pointwise to a function , then is continuous and converges to in the compact-open topology.
References
- 1 J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
Title | equicontinuous |
---|---|
Canonical name | Equicontinuous |
Date of creation | 2013-03-22 18:38:10 |
Last modified on | 2013-03-22 18:38:10 |
Owner | asteroid (17536) |
Last modified by | asteroid (17536) |
Numerical id | 6 |
Author | asteroid (17536) |
Entry type | Definition |
Classification | msc 54E35 |
Classification | msc 54C35 |
Synonym | equicontinuity |
Related topic | Equicontinuous2 |