Smoothness condition

In the mathematical theory of normalized spaces , certain classes of such spaces are defined by properties of the norm . Here one considers smoothness conditions , i.e. the differentiability properties of the norm . In addition, there are a number of convexity conditions that are related to the smoothness conditions via the dual spaces .

Smoothness conditions

Let it be a standardized space with the unit sphere . One can show that for the limit values ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle S_{X}:=\{x\in X;\,\|x\|=1\}}$ ${\displaystyle x,y\in S_{X}}$

${\displaystyle G_{-}(x,y):=\lim _{t\nearrow 0}{\frac {\|x+ty\|-1}{t}}\quad \quad G_{+}(x,y):=\lim _{t\searrow 0}{\frac {\|x+ty\|-1}{t}}}$

exist and always is. It is said that the norm is differentiable at the point in the direction of Gâteaux if there is equality. The common value is then called ${\displaystyle G_{-}(x,y)\leq G_{+}(x,y)}$ ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle G(x,y)=\lim _{t\rightarrow 0}{\frac {\|x+ty\|-1}{t}}}$

and says the Gâteaux differential exists towards . Classes of standardized rooms are defined by requirements for this limit value. ${\displaystyle x}$ ${\displaystyle y}$

Smooth spaces

→ Main article : Smooth space

The simplest requirement for the limit value to the Gâteaux differential is its existence. We define:

${\displaystyle (X,\|\cdot \|)}$is called smooth if the Gâteaux differential exists for all . ${\displaystyle G(x,y)}$ ${\displaystyle x,y\in S_{X}}$

Evenly smooth spaces

→ Main article : Evenly smooth space

The limit in the definition of smoothness exists for every pair . If one demands uniform convergence here, one obtains a smaller class of normalized spaces: ${\displaystyle G(x,y)}$ ${\displaystyle (x,y)\in S_{X}\times S_{X}}$

${\displaystyle (X,\|\cdot \|)}$is called uniformly smooth if the Gâteaux differential exists uniformly on . ${\displaystyle G(x,y)}$ ${\displaystyle S_{X}\times S_{X}}$

Fréchet-smooth spaces

By restricting the uniformity requirement in the definition of uniform smoothness to the direction variable, one arrives at the following definition:

${\displaystyle (X,\|\cdot \|)}$is called Fréchet-smooth if the Gâteaux differential exists for each equally for . ${\displaystyle G(x,y)}$ ${\displaystyle x\in S_{X}}$ ${\displaystyle y\in S_{X}}$

Evenly gâteaux-smooth spaces

The following class of normalized spaces results if one demands uniformity for the first variable:

${\displaystyle (X,\|\cdot \|)}$is called uniformly Gâteaux-smooth if the Gâteaux differential exists equally for each direction . ${\displaystyle G(x,y)}$ ${\displaystyle y\in S_{X}}$ ${\displaystyle x\in S_{X}}$

Very smooth spaces

If it is smooth, there is exactly one with for each . Thereby an image is defined, called the one spherical image and can be shown by the one that they are related to the. Relative standard topology and the relative weak - * - topology on is continuous. The following definition therefore intensifies the concept of smooth space: ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle x\in S_{X}}$ ${\displaystyle f\in S_{X'}}$ ${\displaystyle \mathrm {Re} f(x)=1}$ ${\displaystyle \sigma :S_{X}\rightarrow S_{X'}}$ ${\displaystyle x\in S_{X}}$ ${\displaystyle f\in S_{X'}}$

A normalized space is called very smooth if it is smooth and the spherical mapping is continuous with respect to the relative norm topology and the relative weak topology . ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle X\setminus \{0\}}$ ${\displaystyle X'\setminus \{0\}}$

The even stronger continuity with regard to the standard topologies leads to the above-mentioned concept of uniformly smooth space.

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:

• US: uniformly smooth
• UG: uniformly Gâteaux smooth
• F: Fréchet smooth
• VS: very smooth

All of the relationships shown here can be found in the textbook by Robert E. Megginson given below.

Relationships with convexity conditions

Let it be a standardized space and its dual space. Then the following statements apply: ${\displaystyle X}$ ${\displaystyle X'}$

• If it is strictly convex , then it is smooth; the converse is generally not true. ${\displaystyle X'}$ ${\displaystyle X}$
• If smooth, then it is strictly convex; the converse is generally not true. ${\displaystyle X'}$ ${\displaystyle X}$
• If it is strongly convex , it is Fréchet-smooth, the converse is generally not true. ${\displaystyle X'}$ ${\displaystyle X}$
• ${\displaystyle X}$is uniformly smooth if and only if is uniformly convex . The roles of and can be reversed. ${\displaystyle X'}$ ${\displaystyle X}$ ${\displaystyle X'}$
• ${\displaystyle X}$is strongly convex if and only if Fréchet is smooth. The roles of and can be reversed. ${\displaystyle X'}$ ${\displaystyle X}$ ${\displaystyle X'}$
• ${\displaystyle X}$is uniformly Gâteaux-smooth if and only if weak * is uniformly convex . ${\displaystyle X'}$

Smoothness module

If it is a standardized space, it is called ${\displaystyle (X,\|\cdot \|)}$

${\displaystyle \rho _{X}:[0,\infty )\rightarrow [0,\infty ),\quad \rho _{X}(t):={\frac {1}{2}}\sup\{\|x+y\|+\|x-y\|-2;\,x,y\in X,\|x\|=1,\|y\|=t\}}$

the smoothness module of . ${\displaystyle (X,\|\cdot \|)}$

The investigation of this function enables further insights into the room classes presented here. For example:

• ${\displaystyle (X,\|\cdot \|)}$   is evenly smooth   . ${\displaystyle \Leftrightarrow }$   ${\displaystyle \lim _{t\searrow 0}{\frac {\rho _{X}(t)}{t}}=0}$

This is used as the definition of uniform smoothness in the Istratescu textbook given below. The estimate applies to the continuity module

${\displaystyle \rho _{X}(t)\geq {\sqrt {1+t^{2}}}-1}$   for any uniformly convex space.

In the extreme case, a characterization of the Hilbert spaces is obtained :

• ${\displaystyle (X,\|\cdot \|)}$   is Hilbert space is a uniformly convex Banach space with   . ${\displaystyle \Leftrightarrow }$   ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle \rho _{X}(t)={\sqrt {1+t^{2}}}-1}$

literature

In the cited textbook by Istratescu there are further smoothness properties that define classes of normalized spaces. Unfortunately, this book contains many errors and is restricted to Banach spaces. Therefore, most of the individual references were based on RE Megginson's textbook, even if the description there is not as extensive.

Individual evidence

1. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , This is not the definition there, but equivalent to it, Corollary 5.4.18
2. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , This is not the definition there, but equivalent to it, sentence 5.5.6
3. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 5.6.1
4. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 5.6.13
5. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 5.6.19
6. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentence 5.4.5
7. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentence 5.4.6
8. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentence 5.6.12
9. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentence 5.5.12
10. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentence 5.6.9
11. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentence 5.6.15
12. ^ Vasile I. Istratescu: Strict Convexity and Complex Strict Convexity, Theory and Applications , Taylor & Francis Inc. (1983), ISBN 0-8247-1796-1 , Definition 2.7.2, only defined there for Banach spaces
13. ^ Vasile I. Istratescu: Strict Convexity and Complex Strict Convexity, Theory and Applications , Taylor & Francis Inc. (1983), ISBN 0-8247-1796-1 , definition 2.7.3
14. ^ Vasile I. Istratescu: Strict Convexity and Complex Strict Convexity, Theory and Applications , Taylor & Francis Inc. (1983), ISBN 0-8247-1796-1 , Corollaries 2.7.9 and 2.7.10