Banach-Mackey theorem

The Banach-Mackey theorem (after Stefan Banach and George Mackey ) is a theorem from the mathematical branch of functional analysis . It makes a statement about the boundedness properties of certain sets in locally convex spaces .

Banach balls

If an absolutely convex subset of a locally convex space is a subspace of , which becomes a normalized space due to the restricted Minkowski functional . If this normalized space is even a Banach space , it is called a Banach sphere . ${\displaystyle B\subset E}$ ${\displaystyle \textstyle E_{B}:=\bigcup _{t>0}tB}$ ${\displaystyle E}$ ${\displaystyle E_{B}}$ ${\displaystyle B}$

• The unit sphere of a normalized space is a Banach sphere if and only if the normalized space is a Banach space. ${\displaystyle E_{B}}$
• In the sequence space of all real sequences is the set of all sequences with for all and for a Banach sphere, because the Minkowki functional of on is equal to the maximum norm . ${\displaystyle \omega }$ ${\displaystyle B}$ ${\displaystyle (x_{i})_{i}}$ ${\displaystyle x_{i}=0}$ ${\displaystyle i>n}$ ${\displaystyle |x_{i}|\leq 1}$ ${\displaystyle i=1,\ldots ,n}$ ${\displaystyle B}$ ${\displaystyle E_{B}\cong \mathbb {R} ^{n}}$
• Every absolutely convex, closed, bounded, sequence-complete subset of a locally convex space is a Banach sphere, in particular compact , absolutely convex sets are Banach spheres.
• Banach spheres can be used to characterize ultrabornological spaces (see there).

The Banach-Mackey Theorem

A subset of a locally convex space is called weakly bounded if the image is bounded under every continuous, linear functional . is called strongly bounded if for all subsets of the dual space for which holds for all . ${\displaystyle B\subset E}$ ${\displaystyle B}$ ${\displaystyle \sup\{|f(x)|;\,x\in B,f\in M\}<\infty }$ ${\displaystyle M\subset E^{'}}$ ${\displaystyle \sup\{|f(x)|;\,f\in M\}<\infty }$ ${\displaystyle x\in E}$

By taking one-element sets for the sets in the above definition, one sees that strongly-bounded sets are weak-bounded. The opposite applies: ${\displaystyle M}$

• Banach-Mackey theorem : Every weakly-bounded Banach sphere in a locally convex space is strongly-bounded.

Applications

• The set of Mackey can be derived from the set of Banach Mackey.
• If every absolutely convex, closed and limited quantity in a quasi-tunnelled space is a Banach sphere, then this space is already tunnelled . In particular, all sequential, quasitonnelierte rooms are already barreled.

Individual evidence

1. ^ R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8 , Corollar 23.14
2. ^ R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8 , sentence 23.12
3. ^ R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8 , sentence 23.15
4. ^ R. Meise, D. Vogt: Introduction to Functional Analysis , Vieweg, 1992 ISBN 3-528-07262-8 , sentence 23.20 + 23.21