Theorem of Banach stone house

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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:

  1. The operator norm sequence is a bounded sequence within the real numbers .
  2. 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

the uniform limitation

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

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

Individual evidence

  1. ^ Harry Hochstadt: Eduard Helly, father of the Hahn-Banach theorem . In: The Mathematical Intelligencer , Vol. 2, 1980, No. 3, pp. 123-125
  2. Hirzebruch, Scharlau: Introduction to Functional Analysis. 1971, p. 22