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In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, 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.

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• Uniform boundedness principle
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• In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, 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.
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• A.I.
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• b/b015200
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• Shtern
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• Suppose that for every x in the Banach space X, one has: : For every integer , let : The set is a closed set and by the assumption, : By the Baire category theorem for the non-empty complete metric space X, there exists m such that has non-empty interior, i.e., there exist and
• T ∈ F
• . One has that: : Taking the supremum over u in the unit ball of X and over T &isin; F it follows that :
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