The principle of uniform boundedness

The principle of uniform boundedness or the Banach – Steinhaus theorem is a fundamental result of functional analysis . The theorem states that pointwise and uniform boundedness are equivalent for families of continuous linear operators defined on a Banach space .

Content

1 History
2 Wording
3 Consequences
4 Variations and generalizations
5 References

History

The theorem was proved by Banach and Steinhaus and independently by Hans Khan .

Wording

Let be $X$ ${\ displaystyle X}$ Banach space $Y$ ${\ displaystyle Y}$ normalized vector space and $F$ ${\ displaystyle F}$ family of linear continuous operators from $X$ ${\ displaystyle X}$ at $Y$ ${\ displaystyle Y}$ . Suppose for any $x\in X$ ${\ displaystyle x \ in X}$ performed

\sup \nolimits _{T\in F}\|T(x)\|_{Y}<\infty ,

{\ displaystyle \ sup \ nolimits _ {T \ in F} \ | T (x) \ | _ {Y} <\ infty,}

then

\sup \nolimits _{T\in F,\|x\|=1}\|T(x)\|_{Y}=\sup \nolimits _{T\in F}\|T\|_{B(X,Y)}<\infty .

{\ displaystyle \ sup \ nolimits _ {T \ in F, \ | x \ | = 1} \ | T (x) \ | _ {Y} = \ sup \ nolimits _ {T \ in F} \ | T \ | _ {B (X, Y)} <\ infty.}

Consequences

If a sequence of bounded operators on a Banach space converges pointwise, then its pointwise limit is a bounded operator.

Variations and generalizations

Barrel space is the most common type of space in which the principle of uniform boundedness is fulfilled.

The boundedness principle holds for mapping families from $X$ ${\ displaystyle X}$ at $Y$ ${\ displaystyle Y}$ if $X$ ${\ displaystyle X}$ is Baire space and $Y$ ${\ displaystyle Y}$ - locally convex space .

References

Banach, Stefan & Steinhaus, Hugo (1927), " Sur le principe de la condensation de singularités ", Fundamenta Mathematicae T. 9: 50–61 , < http://matwbn.icm.edu.pl/ksiazki/fm/fm9 /fm918.pdf > (fr.)
Bourbaki, Nicolas (1987), Topological vector spaces , Elements of mathematics, Springer, ISBN 978-3-540-42338-6
Dieudonné, Jean (1970), Treatise on analysis, Volume 2 , Academic Press .
Rudin, Walter (1966), Real and complex analysis , McGraw-Hill .
Shtern, AI (2001), "Banach – Steinhaus theorem" , in Hazewinkel, Michiel, Encyclopedia of Mathematics , Springer , ISBN 978-1-55608-010-4 .
Sokal, Alan (2011), " A really simple elementary proof of the uniform boundedness theorem ", Amer. Math. Monthly T. 118: 450-452 , DOI 10.4169 / amer.math.monthly.118.05.450 .
Weinberg M.M. Functional Analysis. - M .: Education, 1979. - 128 p.