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Banach-Steinhaus Theorem


The Banach-Steinhaus theorem is a result in the field of functional analysis which relates the "size" of a certain subset of points defined relative to a family of linear mappings between topological vector spaces to a certain continuity property of the maps involved.

More precisely, suppose that X and Y are topological vector spaces, that Gamma={Lambda_alpha}_(alpha in A) is a collection of continuous linear maps from X into Y, and that B denotes the set of all points x in X whose orbits Gamma(x)={Lambda_alphax:Lambda_alpha in Gamma} are bounded in Y. The Banach-Steinhaus theorem says that if B is of second category in X, then it necessarily follows that B=X and that the collection Gamma is equicontinuous.

The statement of the Banach-Steinhaus theorem is often given in various forms, some apparently differing from the above. As a result, various corollaries thereof are sometimes considered part of the actual theorem. One such example is the identification of the Banach-Steinhaus theorem with the so-called uniform boundedness principle, which states that any family of continuous linear operators between Banach spaces is uniformly bounded provided that it is bounded pointwise. This result is actually a corollary of the above-stated version of the Banach-Steinhaus theorem along with the observation that in the above-described framework, an equicontinuous family Gamma necessarily satisfies a uniform boundedness property in which every bounded subset E of X implies the existence of a bounded subset F of Y satisfying Lambda_alpha(E) subset F for every Lambda_alpha in Gamma.


See also

Equicontinuous, Second Category, Uniform Boundedness Principle

This entry contributed by Christopher Stover

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References

Rudin, W. Functional Analysis. New York: McGraw-Hill, 1991.

Cite this as:

Stover, Christopher. "Banach-Steinhaus Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Banach-SteinhausTheorem.html

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