Uniform Boundedness Principle - Generalizations

Generalizations

The least restrictive setting for the uniform boundedness principle is a barrelled space where the following generalized version of the theorem holds (Bourbaki 1987, Theorem III.2.1):

• 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).

Alternatively, the statement also holds whenever X is a Baire space and Y is a locally convex space (Shtern 2001).

Dieudonné (1970) proves a weaker form of this theorem with Fréchet spaces rather than the usual Banach spaces. Specifically,

• Let X be a Fréchet space, Y a normed space, and H a set of continuous linear mappings of X into Y. If for every x∈X, then the family H is equicontinuous.