Local theory of finite dimensional normed spaces: Type and cotype

2021 ◽  
pp. 257-364
2004 ◽  
Vol 69 (2) ◽  
pp. 289-295 ◽  
Author(s):  
M. I. Ostrovskii

One of the important problems of the local theory of Banach Spaces can be stated in the following way. We consider a condition on finite sets in normed spaces that makes sense for a finite set any cardinality. Suppose that the condition is such that to each set satisfying it there corresponds a constant describing “how well” the set satisfies the condition.The problem is:Suppose that a normed space X has a set of large cardinality satisfying the condition with “poor” constant. Does there exist in X a set of smaller cardinality satisfying the condition with a better constant?In the paper this problem is studied for conditions associated with one of R.C. James's characterisations of superreflexivity.


Author(s):  
Paweł Wójcik

AbstractWe observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space $$\ell _2$$ ℓ 2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.


Mathematika ◽  
2013 ◽  
Vol 59 (2) ◽  
pp. 477-492 ◽  
Author(s):  
Horst Martini ◽  
Christian Richter ◽  
Margarita Spirova

Sign in / Sign up

Export Citation Format

Share Document