Real line and complex planeEdit
be a family of functions indexed by , where is an arbitrary set and is the set of real or complex numbers. We call uniformly bounded if there exists a real number such that
In general let be a metric space with metric , then the set
is called uniformly bounded if there exists an element from and a real number such that
- Every uniformly convergent sequence of bounded functions is uniformly bounded.
- The family of functions defined for real with traveling through the integers, is uniformly bounded by 1.
- The family of derivatives of the above family, is not uniformly bounded. Each is bounded by but there is no real number such that for all integers
- Ma, Tsoy-Wo (2002). Banach-Hilbert spaces, vector measures, group representations. World Scientific. p. 620pp. ISBN 981-238-038-8.