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Uniform Boundedness Principle


A "pointwise-bounded" family of continuous linear operators from a Banach space to a normed space is "uniformly bounded." Symbolically, if sup||T_i(x)|| is finite for each x in the unit ball, then sup||T_i|| 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 in X,sup_(A in A)||Ax||<infty, then sup_(A in A)||A||<infty.


See also

Banach-Steinhaus Theorem

Portions of this entry contributed by Mohammad Sal Moslehian

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References

Conway, J. B. A Course in Functional Analysis. New York: Springer-Verlag, 1990.Zeidler, E. Applied Functional Analysis: Applications to Mathematical Physics. New York: Springer-Verlag, 1995.

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Uniform Boundedness Principle

Cite this as:

Moslehian, Mohammad Sal and Weisstein, Eric W. "Uniform Boundedness Principle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/UniformBoundednessPrinciple.html

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