Given a barrelled space " X " and a locally convex space " Y ", then any family of pointwise bounded continuous linear mappings from " X " to " Y " is equicontinuous ( even uniformly equicontinuous ).
22.
For example, if the sequence consists of differentiable functions or functions with some regularity ( e . g ., the functions are solutions of a differential equation ), then the mean value theorem or some other kinds of estimates can be used to show the sequence is equicontinuous.
23.
Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of " C " ( " X " ), the space of continuous functions on a compact Hausdorff space " X ", is compact if and only if it is closed, pointwise bounded and equicontinuous.
24.
*When is Cauchy's formula ) its derivative has modulus bounded by } } in the smaller disk " B " ( " z " 0, ) . } } If a family of holomorphic functions on is bounded by on, it follows that the family of restrictions to is equicontinuous on.
25.
:: : OK, then the topology should be the topology of uniform convergence; but in fact on equicontinuous families it coincides with the topology of pointwise convergence ( the one induced by the product topology; incidentally, this fact yields to a proof of AA theorem, via Tychonoff's theorem )-- a 22 : 23, 21 May 2012 ( UTC)
