Theorem of Banach stone house
The Banach-Steinhaus theorem is one of the fundamental results of functional analysis , one of the branches of mathematics . In the literature, three different but related sentences are often referred to as the Banach-Steinhaus sentence . The most abstract version is also known as the principle of uniform limitation , which in turn follows from Osgood's theorem. The other two versions are consequences of this. Like the theorem on open mapping , these theorems are based on Baire's famous category theorem . Together with Hahn-Banach's theorem , all these theorems are considered cornerstones of the field.
Hugo Steinhaus and Stefan Banach published the theorem in 1927. However, it was also independently proven by Hans Hahn . But it can be found mainly in 1912 with Eduard Helly .
Theorem of Banach stone house
Let be and Banach spaces and with a sequence of continuous linear operators.
Then: converges pointwise to a continuous linear operator if and only if both of the following conditions are met:
- The operator norm sequence is a bounded sequence within the real numbers .
- It exists in a dense subset such that for each the sequence converges within .
Banach stone house theorem (variant)
Let be a Banach space , a normalized space and with a sequence of continuous linear operators .
Then the following applies: If converges pointwise , then defines a continuous linear operator and it applies
Principle of uniform limitation
Let be a Banach space , a normalized vector space and a family of continuous, linear operators from to .
Then it follows from the pointwise restriction
- for all
Proof of the principle of uniform limitedness
Put for . These quantities are obviously closed and applies after acceptance . As a Banach space is completely metrizable and thus a Baire space (see Baire's category theorem ), so it must not be that all are lean. So there is one so that is not lean. Because of the isolation, this means that it is sealed somewhere. That is, there is one and one such that . For everyone and with now applies
- .
Hence for all , so that there is an equal bound on the set .
Remarks
- Point-by-point convergence of operators is, in contrast to weak convergence, also referred to as strong convergence and should not be confused with the even stronger norm convergence .
- The completeness of is an essential prerequisite in the above variant in order to be able to apply the principle of uniform restriction. If, as in the main version, one only assumes point-wise convergence on a dense subset , the boundedness of the sequence of operator norms must also be assumed.
- It is easiest to follow the main version above with the help of the variant and this in turn from the principle of uniform restriction.
Inferences
- Every weakly convergent sequence of a normalized vector space is bounded.
Generalizations
For linear operators on barreled spaces
The general form of the sentence applies to barreled rooms :
If a barreled space is a locally convex space , then the following applies: Every family of pointwise restricted, continuous, linear operators from to is uniformly continuous (even uniformly uniformly continuous ).
The barrel-shaped spaces are precisely those locally convex spaces in which the Banach-Steinhaus theorem applies.
For continuous real-valued functions on complete metric spaces
With Hirzebruch - Scharlau one finds the following very general version of the boundedness principle in the context of the complete metric spaces :
Given a complete metric space and a family of continuous real-valued functions
- ,
which is evenly limited upwards at points:
- .
Then there is in a non-empty open subset such that the family of on limited functions even uniformly bounded up is so the condition
enough.
For continuous real-valued functions on topological spaces
In addition, there is a very far-reaching generalization for continuous real-valued functions on any topological space. This is the content of Osgood's theorem in functional analysis .
literature
- Stefan Banach, Hugo Steinhaus. " Sur le principle de la condensation de singularités (PDF; 568 kB). Fundamenta Mathematicae , 9 50-61, 1927.
- Harro Heuser : Functional Analysis. Theory and application (= mathematical guidelines ). 3rd, revised edition. Teubner Verlag, Stuttgart 1992, ISBN 3-519-22206-X .
- Friedrich Hirzebruch , Winfried Scharlau : Introduction to functional analysis (= BI university pocket books . Volume 296 ). Bibliographisches Institut, Mannheim [u. a.] 1971, ISBN 3-411-00296-4 ( MR0463864 ).
- Ronald Larsen: Functional Analysis. An Introduction (= Pure and Applied Mathematics . Volume 15 ). Marcel Dekker, New York 1973, ISBN 0-8247-6042-5 ( MR0461069 ).
- Kōsaku Yosida : Functional Analysis (= Basic Teachings of the Mathematical Sciences . Volume 123 ). 6th edition. Springer Verlag, Berlin / Heidelberg / New York 1980, ISBN 3-540-10210-8 , Section II.1 The uniform boundedness theorem and the resonance theorem , p. 68 f .