Uniform boundedness principle

The Banach - Steinhaus is one of the fundamental results in functional analysis, one of the branches of mathematics. In the literature, three different, but related sets are called Banach - Steinhaus often. The abstract version is also known as the principle of uniform boundedness, which follows in turn from the set of Osgood. The other two versions are consequences from this. Just as the phrase on the open figure these rates are based on the famous set of Baire category. Together with the Hahn- Banach all these sentences are considered cornerstones of the region.

Hugo Steinhaus and Stefan Banach released the sentence 1927. However, he was also demonstrated by Hans Hahn regardless.

• 4.1 Consequences

Banach - Steinhaus

Let and be Banach spaces and with a sequence of continuous linear operators.

Then: converges pointwise to a continuous linear operator if and only if the following two conditions are met:

Banach - Steinhaus ( variant)

Be a Banach space and a normed space with a sequence of continuous linear operators.

Then: If converges pointwise, so defines a continuous linear operator and it is

Principle of uniform boundedness

Be a Banach space is a normed vector space and a family of continuous linear operators from to.

Then it follows from the pointwise boundedness

The uniform boundedness

Proof of principle of uniform boundedness

Using the Baire'schen category set:

Comments

• Pointwise convergence of operators is called in contrast to the weak convergence and strong convergence as and should not be confused with the even stronger norm convergence.
• The completeness of is an essential requirement in the above variant, in order to apply the principle of uniform boundedness can. If one assumes as in the main version only pointwise convergence on a dense subset, the boundedness of the sequence of operator norms must also be provided.
• The easiest way to follow the above main version using the version and this in turn from the principle of uniform Beschränktkeit.

Conclusions

• Every weakly convergent sequence of a normed vector space is limited.

Generalization

The general form of the theorem is true for barreled spaces:

If a barreled space, a locally convex space, then: Each family pointwise bounded, continuous, linear operators from to is equicontinuous ( even uniformly equicontinuous ).

The barreled spaces are precisely those locally convex spaces in which the theorem of Banach - Steinhaus.