Also called the uniform boundedness principle, it states that a family of bounded linear operators on a Banach space that is pointwise bounded is in fact uniformly bounded in operator norm. Named for Stefan Banach and Hugo Steinhaus, it is one of the cornerstone theorems of functional analysis.
Facts
StatementFor a family of continuous linear operators whose domain is a Banach space, pointwise boundedness of the family is equivalent to uniform boundedness in operator norm. 1 Connections
Sources
1. Banach-Steinhaus Theorem (Wikipedia)
Wikimedia Foundationlead section, third sentence (article titled Uniform boundedness principle)
it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm
lead section, history sentence
The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus, but it was also proven independently by Hans Hahn.
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