Convexity condition

In the mathematical theory of normalized spaces , certain classes of normalized spaces are defined by properties of the unit sphere . Here we consider convexity conditions that sharpen the convexity of the unit sphere. Its origins go back to the 1930s, here especially James A. Clarkson as well as Mark G. Kerin and Naum I. Achijeser should be mentioned for basic concept formation.

In addition, there are a number of smoothness conditions that investigate the differentiability properties of the standard . There is a close relationship between convexity conditions and smoothness conditions via the dual spaces .

Convexity Conditions

In the case of the sum norm (right) it is equal to 1, in the case of the Euclidean norm (left) the convexity inequality is strict. ${\displaystyle \|\textstyle {\frac {1}{2}}(e_{1}+e_{2})\|}$

It is a standardized space. In formulas, the convexity means the unit sphere ${\displaystyle (X,\|\cdot \|)}$

For every two vectors with and applies . ${\displaystyle x,y\in X}$ ${\displaystyle \|x\|\leq 1,\,\|y\|\leq 1}$ ${\displaystyle t\in [0,1]}$ ${\displaystyle \|tx+(1-t)y\|\leq 1}$

This does not rule out that in many cases equality also exists for different vectors and for , as is the case, for example, with a square as a unit sphere. By excluding this or setting even more stringent conditions, one arrives at the room classes presented below. For a simpler formulation, a standardized space is always included ${\displaystyle 0<t<1}$ ${\displaystyle (X,\|\cdot \|)}$

Unit sphere ${\displaystyle B_{X}:=\{x\in X;\,\|x\|\leq 1\}}$

and unity sphere . ${\displaystyle S_{X}:=\{x\in X;\,\|x\|=1\}}$

In order to rule out the trivial, do not be the null space . be the dual space with the dual space norm defined by . ${\displaystyle X}$ ${\displaystyle X^{*}}$ ${\displaystyle \|f\|:=\sup\{|f(x)|;\,x\in B_{X}\}}$

Strictly convex spaces

→ Main article : Strictly convex space

If one wants to exclude the equality in the convexity formula as far as possible, i.e. to ensure that the unit sphere does not contain any lines, this leads to the following definition:

${\displaystyle (X,\|\cdot \|)}$ is called strictly convex if:

Are different and is , so is . ${\displaystyle x,y\in S_{X}}$ ${\displaystyle 0<t<1}$ ${\displaystyle \|tx+(1-t)y\|<1}$

Uniformly convex spaces

→ Main article : Uniformly convex space

By controlling how well the inequality in the definition of strictly convex space is satisfied, one arrives at the following concept:

${\displaystyle (X,\|\cdot \|)}$ is called uniformly convex if:

For each   is   . ${\displaystyle 0<\varepsilon \leq 2}$ ${\displaystyle \inf\{1-{\frac {1}{2}}\|x+y\|;\,x,y\in S_{X},\|x-y\|\geq \varepsilon \}>0}$

Locally uniformly convex spaces

The condition in the definition of uniform convexity applies equally to all . By holding one vector and only building the infimum over the other, you get the following local version: ${\displaystyle x,y}$

${\displaystyle (X,\|\cdot \|)}$ is called locally uniformly convex if:

For each     and every one     is   . ${\displaystyle 0<\varepsilon \leq 2}$ ${\displaystyle x\in S_{X}}$ ${\displaystyle \inf\{1-{\frac {1}{2}}\|x+y\|;\,y\in S_{X},\|x-y\|\geq \varepsilon \}>0}$

Weakly evenly convex spaces

In the definition of uniform convexity, the condition in the set over which the infimum is formed can be too weakened, where is, i.e. comes from the unit sphere of the dual space. ${\displaystyle \|x-y\|\geq \varepsilon }$ ${\displaystyle |f(x-y)|\geq \varepsilon }$ ${\displaystyle f\in S_{X^{*}}}$

${\displaystyle (X,\|\cdot \|)}$ is called weakly uniformly convex if it holds

For each     and     is   . ${\displaystyle 0<\varepsilon \leq 2}$ ${\displaystyle f\in S_{X^{*}}}$ ${\displaystyle \inf\{1-{\frac {1}{2}}\|x+y\|;\,x,y\in S_{X},|f(x-y)|\geq \varepsilon \}>0}$

Locally weakly evenly convex spaces

The condition in the definition of the weakly uniform convexity can again be weakened to a local version:

${\displaystyle (X,\|\cdot \|)}$ is called locally weakly uniformly convex if:

For each   ,   and     is   . ${\displaystyle 0<\varepsilon \leq 2}$ ${\displaystyle x\in S_{X}}$ ${\displaystyle f\in S_{X^{*}}}$ ${\displaystyle \inf\{1-{\frac {1}{2}}\|x+y\|;\,y\in S_{X},|f(x-y)|\geq \varepsilon \}>0}$

Locally evenly center-convex spaces

From the uniform convexity it follows that for two sequences and in , for which the norm of the sequence of the center points converges to 1, must hold. This condition can be weakened by the requirement that the sequence of the center points must actually converge to an element of the unit sphere: ${\displaystyle (x_{n})_{n}}$ ${\displaystyle (y_{n})_{n}}$ ${\displaystyle S_{X}}$ ${\displaystyle \textstyle {\frac {1}{2}}(x_{n}+y_{n})}$ ${\displaystyle \|x_{n}-y_{n}\|\rightarrow 0}$

${\displaystyle (X,\|\cdot \|)}$ is called locally uniformly center-point convex if:

If   and     sequences   converge in   and  , then we have   . ${\displaystyle (x_{n})_{n}}$ ${\displaystyle (y_{n})_{n}}$ ${\displaystyle S_{X}}$ ${\displaystyle \textstyle {\frac {1}{2}}(x_{n}+y_{n})\rightarrow z\in S_{X}}$ ${\displaystyle \|x_{n}-y_{n}\|\rightarrow 0}$

Uniformly convex spaces in every direction

→ Main article : UCED area

A further generalization arises when one can only infer if the differences all have the same direction. ${\displaystyle \|x_{n}-y_{n}\|\rightarrow 0}$ ${\displaystyle x_{n}-y_{n}}$

${\displaystyle (X,\|\cdot \|)}$is called uniformly convex in the direction if: ${\displaystyle z\in X\setminus \{0\}}$

Are     and     consequences in     and converge     and is   for all , then applies   . ${\displaystyle (x_{n})_{n}}$ ${\displaystyle (y_{n})_{n}}$ ${\displaystyle S_{X}}$ ${\displaystyle \textstyle {\frac {1}{2}}(x_{n}+y_{n})\rightarrow z\in S_{X}}$ ${\displaystyle x_{n}-y_{n}\in \mathbb {R} z}$ ${\displaystyle n}$ ${\displaystyle \|x_{n}-y_{n}\|\rightarrow 0}$

${\displaystyle (X,\|\cdot \|)}$is called uniformly convex in every direction or UCED space if is uniformly convex in direction for all . ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle z\in X\setminus \{0\}}$ ${\displaystyle z}$

Strongly convex spaces

The non-empty intersection of sphere and convex set becomes arbitrarily small.

→ Main article : Strongly convex space

In order to achieve, as in the tightening of convexity to strict convexity, that the unit sphere does not contain any segments, one can consider the intersections of the spheres with radius with a convex set and demand that the diameter of the non-empty intersections approaches 0 with decreasing radius. ${\displaystyle rB_{X}}$ ${\displaystyle r}$ ${\displaystyle C}$ ${\displaystyle \mathrm {diam} (rB_{X}\cap C)}$

${\displaystyle (X,\|\cdot \|)}$ is called strongly convex if:

For every convex set   we have     for   . ${\displaystyle C\subset X}$ ${\displaystyle \mathrm {diam} (rB_{X}\cap C)\rightarrow 0}$ ${\displaystyle r\searrow \inf\{t>0;\,tB_{X}\cap C\not =\emptyset \}}$

Weak * evenly convex spaces

The above weakening of the uniform convexity using the weak topology can be formulated on the dual space with the weak - * - topology :

The dual space is called weak * uniformly convex if: ${\displaystyle (X^{*},\|\cdot \|)}$

For each     and     is   . ${\displaystyle 0<\varepsilon \leq 2}$ ${\displaystyle x\in S_{X}}$ ${\displaystyle \inf\{1-{\frac {1}{2}}\|f+g\|;\,f,g\in S_{X^{*}},|(f-g)x|\geq \varepsilon \}>0}$

Overview

Relationships between the room classes

This diagram gives an overview of the relationships between the room classes, with the class of interior product rooms being the most special. An arrow from one class to the other means that every normalized space of the first class also belongs to the second. The reflexivity of a normalized space means that the completion is a reflexive space . Note that with the exception of reflexivity and, of course, the lowest property of being a normalized space, any of the properties can be lost in the transition to an equivalent norm . The following standard abbreviations, some of which go back to the corresponding English names, were used:

• UR: uniformly convex (uniformly rotund)
• LUR: locally uniformly convex (locally uniformly rotund)
• wUR: weakly uniformly convex (weakly uniformly rotund)
• MLUR: locally uniformly center convex (midpoint locally uniformly rotund)
• UCED: uniformly convex in each direction
• wLUR: weakly locally uniformly convex (weakly locally uniformly rotund)
• H: Radon-Riesz property (no English abbreviation)

Dual spaces

Many of the convexity conditions presented here correspond to smoothness conditions on the dual space. The relationships that apply here are summarized in the article on smoothness conditions .

Convexity module

For a normalized space the figure is called ${\displaystyle (X,\|\cdot \|)}$

${\displaystyle \delta _{X}:[0,2]\rightarrow [0,1],\delta _{X}(t):=\inf\{1-{\frac {1}{2}}\|x+y\|;\,x,y\in E,\|x\|\leq 1,\|y\|\leq 1,\|x-y\|=t\}}$

the convexity module . This is a monotonically increasing function that has the value 0 in 0, even the mapping is monotonically increasing. This allows two spaces to be compared with regard to their convexity properties; one can call a room more convex than a room if for all . ${\displaystyle t\mapsto \delta _{X}(t)/t}$ ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle (Y,\|\cdot \|)}$ ${\displaystyle \delta _{X}(t)\geq \delta _{Y}(t)}$ ${\displaystyle t\in [0,2]}$

A normalized space is uniformly convex if and only if for all . ${\displaystyle X}$ ${\displaystyle \delta _{X}(t)>0}$ ${\displaystyle t>0}$

Obviously, the following applies to the sequence space of the null sequences with the supreme norm ${\displaystyle c_{0}}$

${\displaystyle \delta _{c_{0}}(t)=0}$   for all , ${\displaystyle t\in [0,2]}$

because for and everyone is ${\displaystyle e_{1}:=(1,0,0,\ldots ),e_{2}:=(0,1,0,0,\ldots )\in c_{0}}$ ${\displaystyle t\in [0,2]}$

${\displaystyle \textstyle \|(e_{1}\pm {\frac {t}{2}}e_{2})\|=\|(1,\pm {\frac {t}{2}},0,\ldots )\|=1,\,\,\|(e_{1}+{\frac {t}{2}}e_{2})-(e_{1}-{\frac {t}{2}}e_{2})\|=\|(0,t,0,0,\ldots )\|=t}$ and therefore

${\displaystyle \textstyle \delta _{c_{0}}(t)\leq 1-{\frac {1}{2}}\|(e_{1}+{\frac {t}{2}}e_{2})+(e_{1}-{\frac {t}{2}}e_{2})\|=1-{\frac {1}{2}}\|(2,0,0,\ldots )\|=0}$.

For a Hilbert space it follows using the parallelogram equation ${\displaystyle H}$

${\displaystyle \delta _{H}(t)=1-{\sqrt {1-\left({\frac {t}{2}}\right)^{2}}}}$   for all ${\displaystyle t\in [0,2]}$

and it applies

${\displaystyle \delta _{X}(t)\leq \delta _{H}(t)}$for all evenly convex spaces . In this sense, the Hilbert rooms are the most "convex" rooms. ${\displaystyle X}$

literature

The textbook by Istratescu given below is specifically devoted to this topic and contains further generalizations and many examples. Unfortunately, this book has a lot of mistakes, even in definitions and sentence formulations, which makes it difficult to get started, and it is unnecessarily limited to Banach spaces. In this respect, the textbook by Robert E. Megginson, to which most of the individual proofs refer, is much better and somewhat more recent, and the proof is much more detailed. This is where the basis for the above overview comes from.

Individual evidence

1. ↑ James A. Clarkson: Uniformly convex spaces , Transactions American Mathematical Society (1936), Volume 40, pages 396-414
2. ↑ Naum I. Akhiezer, Mark. G. Kerin: О некоторых вопросах теории моментов (On some questions about moment theory), Charkow (1938), English translation in Translations of Mathematical Monographs, Volume 2, American Mathematical Society, Providence (1962)
3. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 5.1.1
4. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 5.2.1
5. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 5.3.2
6. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 5.3.8
7. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 5.3.13
8. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 5.3.25
9. ↑ Vasile I. Istratescu: Strict Convexity and Complex Strict Convexity, Theory and Applications , Taylor & Francis Inc. (1983), ISBN 0-8247-1796-1 , Definition 2.6.9, the formulation chosen here is according to Theorem 2.6.33 (2) equivalent to this.
10. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 5.3.15
11. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 5.3.11
12. ^ Joseph Diestel: Sequences and Series in Banach Spaces. 1984, ISBN 0-387-90859-5 , Chapter VIII, page 125
13. ^ Vasile I. Istratescu: Strict Convexity and Complex Strict Convexity, Theory and Applications , Taylor & Francis Inc. (1983), ISBN 0-8247-1796-1 , definition 2.7.7
14. ^ G. Nordlander: The modulus of convexity in normed spaces , Arkiv för Math. (1960), Volume 4, pages 15-17