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:
Definition: absolute p-convex[edit | edit source]
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 .
Definition: absolute p-convex hull[edit | edit source]
The absolutely -convex hull of the set (label: )
is the section over all absolutely -convex sets containing .
Lemma: Display of the absolutely p-convex hull[edit | edit source]
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 .
Proof Part 2.1 - Absolute p-convex[edit | edit source]
Let now have the following representations:
- with
- with .
Now we have to show that the absolute -convex combination is an element of , i.e.
proof-part-2.2-absolutely-p-convex[edit | edit source]
is absolutely -convex, because it holds with :
This gives:
Proof Part 2.3 - Zero Vector[edit | edit source]
, because it holds with and any gets .
We now show that the absolutely -convex hull is contained in every absolutely -convex superset of .
Proof Part 3.1 - Induction over Number of Summands[edit | edit source]
Now let us show inductively via the number of summands that every element of the form
in a given absolutely -convex set is contained.
Proof Part 3.2 - Induction Start[edit | edit source]
For , the assertion follows via the definition of an absolutely -convex set .
Proof Part 3.3 - Induction Precondition[edit | edit source]
Now let the condition for hold, i.e.:
Proof Part 3.4 - Induction Step[edit | edit source]
For , the assertion follows as follows:
Let and with for all . is now to be proved.
Proof Part 3.5 - Induction Step[edit | edit source]
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 .
Proof Part 3.8 - Induction Step[edit | edit source]
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.
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