For -norms are a generalization of norms. The definition requires the notion of (absolute) -convex hull (see Köthe 1966[1]).
Let be a subset of a vector space and , then is called -convex if fulfills the following property:
Let be a subset of a vector space and , then is said to be absolutely -convex if fulfills the following property:
The -convex hull of the set (label: ) is the intersection over all -convex sets containing .
The absolutely -convex hull of the set (label: )
is the section over all absolutely -convex sets containing .
Let be a subset of a vector space over the body and , then the absolute -convex hull of can be written as follows:
3 subassertions are shown, where (1) and (2) gives and (3) gives the subset relation .
- (Proof part 1) ,
- (Proof part 2) is absolutely -convex and.
- (Proof part 3) is contained in any absolutely -convex set .
, because
Now let and be given. One must show that .
Let now have the following representations:
- with
- with .
Now we have to show that the absolute -convex combination is an element of , i.e.
is absolutely -convex, because it holds with :
This gives:
, because it holds with and any gets .
We now show that the absolutely -convex hull is contained in every absolutely -convex superset of .
Now let us show inductively via the number of summands that every element of the form
in a given absolutely -convex set is contained.
For , the assertion follows via the definition of an absolutely -convex set .
Now let the condition for hold, i.e.:
For , the assertion follows as follows:
Let and with for all . is now to be proved.
If , then there is nothing to show, since then all are for .
Proof Part 3.6 - Constructing a p-convex combination of n summands
[edit | edit source]
We now construct a sum of non-negative summands
Proof part 3.7 - Application of the induction assumption
[edit | edit source]
So let . The inequality
Returns after induction assumption .
Since is absolutely -convex, it follows with
From the proof parts , and together the assertion follows.
Let be a subset of a vector space over the body and , then the -convex hull of can be written as follows:
Transfer the above proof analogously to the -convex hull.
- ↑ Gottfried Köthe (1966) Topological Vector Spaces, 15.10, pp.159-162.
You can display this page as Wiki2Reveal slides
The Wiki2Reveal slides were created for the Inverse-producing extensions of Topological Algebras' and the Link for the Wiki2Reveal Slides was created with the link generator.