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:
![{\displaystyle \forall _{\displaystyle x,y\in M;\lambda ,\mu \geq 0}:\lambda ^{p}+\mu ^{p}=1\,\Longrightarrow \,\lambda x+\mu y\in M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc98ef924ab5a270017f2c4e75d9aefaa7544132)
Let
be a subset of a vector space
and
, then
is said to be absolutely
-convex if
fulfills the following property:
![{\displaystyle \forall _{\displaystyle x,y\in M}:|\lambda |^{p}+|\mu |^{p}\leq 1\,\Longrightarrow \,\lambda x+\mu y\in M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c3539273e0f92cad0ae6660ec3b0cebc15f2fa)
The
-convex hull of the set
(label:
) is the intersection over all
-convex sets containing
.
![{\displaystyle {\mathcal {C}}_{p}(M):=\displaystyle \bigcap _{\stackrel {{\widetilde {M}}\supseteq M}{{\widetilde {M}}\,p-convex}}{\widetilde {M}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79be7407405432d2f22568725c0a2febc42fe9b2)
The absolutely
-convex hull of the set
(label:
)
is the section over all absolutely
-convex sets containing
.
![{\displaystyle \Gamma _{p}(M):=\displaystyle \bigcap _{\stackrel {{\widetilde {M}}\supseteq M}{{\widetilde {M}}\,absolute\,p-convex}}{\widetilde {M}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/100eb4271755b096b06ccd7d826c0969224548af)
Let
be a subset of a vector space
over the body
and
, then the absolute
-convex hull of
can be written as follows:
![{\displaystyle \Gamma _{p}(M)=\left\{\sum _{j=1}^{n}\alpha _{j}x_{j}\,:\,n\in \mathbb {N} \wedge x_{j}\in M\wedge \sum _{j=1}^{n}|\alpha _{j}|^{p}\leq 1\right\}=:{\widehat {M}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac5d4a72b958453140b83a5346949abb9fa49a62)
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 ![{\displaystyle \displaystyle \sum _{i=1}^{m}|\alpha _{i}|^{p}\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c61289adb789e4e3ea3bc16ec4c7db9a035ece7)
with
.
Now we have to show that the absolute
-convex combination is an element of
, i.e.
is absolutely
-convex, because it holds with
:
![{\displaystyle {\begin{array}{rcl}\sum _{i=1}^{m}|\alpha \alpha _{i}|^{p}+\sum _{j=1}^{n}|\beta \beta _{j}|^{p}&=&|\alpha |^{p}\underbrace {\sum _{i=1}^{m}|\alpha _{i}|^{p}} _{\leq 1}+|\beta |^{p}\underbrace {\sum _{j=1}^{n}|\beta _{j}|^{p}} _{\leq 1}\\&\leq &|\alpha |^{p}+|\beta |^{p}\leq 1.\\\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edc252aaa5b55a76703eb55c51a552e5a1f3e4c7)
This gives:
![{\displaystyle \alpha \cdot x+\beta \cdot y=.\alpha \sum _{i=1}^{m}\alpha _{i}x_{i}+\beta \sum _{j=1}^{n}\beta _{j}y_{j}\in {\widehat {M}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b7c3f4c4d95cd0ce34b20980145277701ca539)
, 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
![{\displaystyle \sum _{j=1}^{n}\alpha _{j}x_{j}{\mbox{ with }}x_{j}\in M{\mbox{ and }}\sum _{j=1}^{n}|\alpha _{j}|^{p}\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba72d5ae48f0f7ca5ced9e33c60c508415a28f9)
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.:
![{\displaystyle \sum _{j=1}^{n}\alpha _{j}x_{j}\in {\widetilde {M}}{\mbox{ with }}x_{j}\in M{\mbox{ and }}\sum _{j=1}^{n}|\alpha _{j}|^{p}\leq 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95078aa88fa061883b32ea76e7839a07d2bc79b3)
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
![{\displaystyle \beta _{j}:={\frac {\alpha _{j}}{\sqrt[{p}]{1-|\alpha _{n+1}|^{p}}}}{\mbox{ with }}\sum _{j=1}^{n}\left|\beta _{j}\right|^{p}\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31aacf3bdd557b313f75904429a2f14e70502bc3)
Proof part 3.7 - Application of the induction assumption
[edit | edit source]
So let
. The inequality
![{\displaystyle \sum _{j=1}^{n}\underbrace {\left|{\frac {\alpha _{j}}{\sqrt[{p}]{1-|\alpha _{n+1}|^{p}}}}\right|^{p}} _{=|\beta _{j}|^{p}}={\frac {1}{1-|\alpha _{n+1}|^{p}}}\cdot \underbrace {\sum _{j=1}^{n}|\alpha _{j}|^{p}} _{\leq 1-|\alpha _{n+1}|^{p}}\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6889d4c5815a45fea377aaa1e54ffec7a4c45b)
Returns after induction assumption
.
Since
is absolutely
-convex, it follows with
![{\displaystyle {\widetilde {M}}\ni \left({\sqrt[{p}]{1-|\alpha _{n+1}|^{p}}}\right)z+\alpha _{n+1}x_{n+1}=\sum _{j=1}^{n}\alpha _{j}x_{j}+\alpha _{n+1}x_{n+1}=.\sum _{j=1}^{n+1}\alpha _{j}x_{j}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/030887298398d1f7b2f37f5de1b433f7794418bd)
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:
![{\displaystyle {\cal {C}}_{p}(M)=\left\{.\sum _{j=1}^{n}\alpha _{j}x_{j}\,:\,n\in \mathbb {N} \wedge x_{j}\in M\wedge \alpha _{j}\in [0,1]\wedge \sum _{j=1}^{n}\alpha _{j}^{p}=1\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc829728e16841baa5685ba6294cdfa1902c1f1d)
Transfer the above proof analogously to the
-convex hull.
- ↑ Gottfried Köthe (1966) Topological Vector Spaces, 15.10, pp.159-162.
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