Local theory of finite dimensional normed spaces: Type and cotype

Finite dimensional characteristics related to superreflexivity of Banach spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700036029 ◽

2004 ◽

Vol 69 (2) ◽

pp. 289-295 ◽

Cited By ~ 1

Author(s):

M. I. Ostrovskii

Keyword(s):

Banach Spaces ◽

Normed Space ◽

Local Theory ◽

Normed Spaces ◽

Finite Sets ◽

Large Cardinality ◽

Finite Dimensional ◽

Finite Set

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.

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When are maps preserving semi-inner products linear?

Aequationes Mathematicae ◽

10.1007/s00010-021-00829-3 ◽

2021 ◽

Author(s):

Paweł Wójcik

Keyword(s):

Hilbert Space ◽

Infinite Dimensions ◽

Normed Spaces ◽

Linear Isometry ◽

Inner Products ◽

Finite Dimensional ◽

Uniformly Smooth ◽

Non Linear ◽

Extra Assumption ◽

Continuous Injection

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.

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