Evenly smooth space

Evenly smooth spaces are examined in the mathematical sub-area of functional analysis. These are standardized spaces , the norm of which fulfills a special smoothness condition. They are closely related to the uniformly convex spaces via a dual space relationship .

Definitions

A normalized space is called smooth if the norm is Gâteaux-differentiable on the unit sphere , that is, if the limit value for each and every one ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle S_{X}:=\{x\in X|\,\|x\|=1\}}$ ${\displaystyle x\in S_{X}}$ ${\displaystyle y\in X}$

${\displaystyle \lim _{t\searrow 0}{\frac {\|x+ty\|-1}{t}}}$

exists. This is exactly the case if for each and ${\displaystyle x\in S_{X}}$ ${\displaystyle y\in X}$

${\displaystyle \lim _{t\searrow 0}{\frac {{\frac {1}{2}}(\|x+ty\|+\|x-ty\|)-1}{t}}=0}$

applies. It is now obvious to apply uniformity conditions to the existence of this limit value. The so-called smoothness modulus of is therefore defined ${\displaystyle X}$

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

and calls the space equally smooth , if ${\displaystyle (X,\|\cdot \|)}$

${\displaystyle \lim _{t\searrow 0}{\frac {\rho _{X}(t)}{t}}=0}$

applies. So that means the expression

${\displaystyle {\frac {{\frac {1}{2}}(\|x+ty\|+\|x-ty\|)-1}{t}}}$

not only for everyone converges against if , but even evenly on . ${\displaystyle (x,y)\in S_{X}\times S_{X}}$ ${\displaystyle 0}$ ${\displaystyle t\searrow 0}$ ${\displaystyle S_{X}\times S_{X}}$

Examples

• Interior product spaces are uniformly smooth, because it is easy to show using the parallelogram equation

${\displaystyle \rho _{X}(t)={\sqrt {1+t^{2}}}-1}$,

from which the uniform smoothness follows.

• The L ^p -spaces for dimensional spaces with positive measure are uniformly smooth if . ${\displaystyle L^{p}(\Omega ,\Sigma ,\mu )}$ ${\displaystyle (\Omega ,\Sigma ,\mu )}$ ${\displaystyle 1<p<\infty }$
• The sequence spaces for are equally smooth. This is a special case of the previous example. The spaces and are not evenly smooth, they are not even smooth. ${\displaystyle \ell ^{p}}$ ${\displaystyle 1<p<\infty }$ ${\displaystyle \ell ^{1}}$ ${\displaystyle \ell ^{\infty }}$
• There is a norm on the sequence space of the null sequences with respect to which this space is smooth but not uniformly smooth. ${\displaystyle c_{0}}$

properties

• Evenly smooth spaces are smooth, because the above definition intensifies an equivalent characterization of the smoothness. The converse also applies to finite-dimensional spaces, but generally not to infinite-dimensional spaces.
• Uniformly smooth Banach spaces are exactly the dual spaces of uniformly convex Banach spaces . In particular, evenly smooth spaces are reflexive , because evenly convex spaces are reflexive according to Milman's theorem.
• Sub- spaces and quoted spaces after closed sub-spaces of evenly smooth spaces are again evenly smooth.
• For smooth areas one has the support picture of each of the uniquely determined support functional maps. This support map is normal - weak- * -continuous. A smooth room is uniformly smooth if and only if the support map is norm-norm-continuous. ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle f:X\setminus \{0\}\rightarrow X^{*}\setminus \{0\},x\mapsto f_{x}}$ ${\displaystyle x\not =0}$ ${\displaystyle f_{x}}$

Individual evidence

1. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentence 5.5.1
2. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 5.5.2
3. ^ Joseph Diestel: Geometry of Banach Spaces - Selected Topics , Springer-Verlag 1975, ISBN 3-540-07402-3 , chap. 3, §4, Proof of Corollary 1 to Theorem 1 '
4. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , Theorem 5.5.16
5. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , example 5.5.15
6. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , Theorem 5.5.12
7. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentences 5.5.20, 5.5.22
8. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentences 5.5.20, 5.5.210