James' theorem

The compactness criterion by James ( by Robert C. James ) is a mathematical sentence from the branch of functional analysis. This theorem characterizes the respect to the weak topology of a Banach space compact sets and has the set of reflexive Banach spaces James result.

A non-empty weakly closed set if and only weakly compact if every continuous linear functional in the dual space takes on this set the maximum amount. Specifically, is this sentence:

Compactness criterion by James: Be a Banach space and a non-empty weakly closed set. Then the following statements are equivalent:

• Is weakly compact.
• For each there is a with.
• For each there is a with.
• For each there is a with.

This is the real vector space created by the restriction on the scalar multiplication. This part of the theorem is only interesting for - Banach spaces. One conclusion from the above sentence is:

Set of James: are equivalent for a Banach space:

• Is reflexive.
• For all there is with so.

This follows immediately from the above compactness criterion, if one uses that a Banach space is reflexive if the unit ball is weakly compact, and that the supremum on the unit sphere is defined as a equal.

Historically, these rates have been demonstrated in the reverse order. First, James 1957 had proved the Reflexivitätskriterium for separable Banach spaces and 1964 for general Banach spaces. Since the reflexivity to the weak compactness of the unit sphere is equivalent, Victor L. Klee had in 1962, this reformulated as a compactness criterion for the unit sphere and suggested that this criterion characterizes any weakly compact sets. This was actually proved in 1964 by RC James.