uniform boundedness principle

Input interpretation

uniform boundedness principle

Alternate name

principle of uniform boundedness

Definition

A "pointwise-bounded" family of continuous linear operators from a Banach space to a normed space is "uniformly bounded." Symbolically, if sup left double bracketing bar T_i(x) right double bracketing bar is finite for each x in the unit ball, then sup left double bracketing bar T_i right double bracketing bar is finite. The theorem is a corollary of the Banach-Steinhaus theorem. Stated another way, let X be a Banach space and Y be a normed space. If A is a collection of bounded linear mappings of X into Y such that for each x element X, sup_(A element A) left double bracketing bar A x right double bracketing bar <∞, then sup_(A element A) left double bracketing bar A right double bracketing bar <∞.
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