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Type and cotype of a Banach space

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In functional analysis, the type and cotype of a Banach space are a classification of Banach spaces through probability theory and a measure, how far a Banach space from a Hilbert space is.

The starting point is the Pythagorean identity for orthogonal vectors in Hilbert spaces

This identity no longer holds in general Banach spaces, however one can introduce a notion of orthogonality probabilistically with the help of Rademacher random variables, for this reason one also speaks of Rademacher type and Rademacher cotype.

The notion of type and cotype was introduced by French mathematician Jean-Pierre Kahane.

Definition

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Let

  • be a Banach space,
  • be a sequence of independent Rademacher random variables, i.e. and for and .

Type

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is of type for if there exist a finite constant such that

for all finite sequences . The sharpest constant is called type constant and denoted as .

Cotype

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is of cotype for if there exist a finite constant such that

respectively

for all finite sequences . The sharpest constant is called cotype constant and denoted as .[1]

Remarks

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By taking the -th resp. -th root one gets the equation for the Bochner norm.

Properties

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  • Every Banach space is of type (follows from the triangle inequality).
  • A Banach space is of type and cotype if and only if the space is also isomorphic to a Hilbert space.

If a Banach space:

  • is of type then it is also type .
  • is of cotype then it is also of cotype .
  • is of type for , then its dual space is of cotype with (conjugate index). Further it holds that [1]

Examples

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  • The spaces for are of type and cotype , this means is of type , is of type and so on.
  • The spaces for are of type and cotype .
  • The space is of type and cotype .[2]

Literature

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  • Li, Daniel; Queffélec, Hervé (2017). Introduction to Banach Spaces: Analysis and Probability. Cambridge Studies in Advanced Mathematics. Cambridge University Press. pp. 159–209. doi:10.1017/CBO9781316675762.009.
  • Joseph Diestel (1984). Sequences and Series in Banach Spaces. Springer New York.
  • Laurent Schwartz (2006). Geometry and Probability in Banach Spaces. Springer Berlin Heidelberg. ISBN 978-3-540-10691-3.
  • Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 23. Berlin, Heidelberg: Springer. doi:10.1007/978-3-642-20212-4_11.

References

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  1. ^ a b Li, Daniel; Queffélec, Hervé (2017). Introduction to Banach Spaces: Analysis and Probability. Cambridge Studies in Advanced Mathematics. Cambridge University Press. pp. 159–209. doi:10.1017/CBO9781316675762.009.
  2. ^ Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 23. Berlin, Heidelberg: Springer. doi:10.1007/978-3-642-20212-4_11.