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Banach–Mazur compactum

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(Redirected from Banach-Mazur distance) Concept in functional analysis Not to be confused with Banach–Mazur game or Banach–Mazur theorem.

In the mathematical study of functional analysis, the Banach–Mazur distance is a way to define a distance on the set Q ( n ) {\displaystyle Q(n)} of n {\displaystyle n} -dimensional normed spaces. With this distance, the set of isometry classes of n {\displaystyle n} -dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum.

Definitions

If X {\displaystyle X} and Y {\displaystyle Y} are two finite-dimensional normed spaces with the same dimension, let GL ( X , Y ) {\displaystyle \operatorname {GL} (X,Y)} denote the collection of all linear isomorphisms T : X Y . {\displaystyle T:X\to Y.} Denote by T {\displaystyle \|T\|} the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors. The Banach–Mazur distance between X {\displaystyle X} and Y {\displaystyle Y} is defined by δ ( X , Y ) = log ( inf { T T 1 : T GL ( X , Y ) } ) . {\displaystyle \delta (X,Y)=\log {\Bigl (}\inf \left\{\left\|T\right\|\left\|T^{-1}\right\|:T\in \operatorname {GL} (X,Y)\right\}{\Bigr )}.}

We have δ ( X , Y ) = 0 {\displaystyle \delta (X,Y)=0} if and only if the spaces X {\displaystyle X} and Y {\displaystyle Y} are isometrically isomorphic. Equipped with the metric δ, the space of isometry classes of n {\displaystyle n} -dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum.

Many authors prefer to work with the multiplicative Banach–Mazur distance d ( X , Y ) := e δ ( X , Y ) = inf { T T 1 : T GL ( X , Y ) } , {\displaystyle d(X,Y):=\mathrm {e} ^{\delta (X,Y)}=\inf \left\{\left\|T\right\|\left\|T^{-1}\right\|:T\in \operatorname {GL} (X,Y)\right\},} for which d ( X , Z ) d ( X , Y ) d ( Y , Z ) {\displaystyle d(X,Z)\leq d(X,Y)\,d(Y,Z)} and d ( X , X ) = 1. {\displaystyle d(X,X)=1.}

Properties

F. John's theorem on the maximal ellipsoid contained in a convex body gives the estimate:

d ( X , n 2 ) n , {\displaystyle d(X,\ell _{n}^{2})\leq {\sqrt {n}},\,}

where n 2 {\displaystyle \ell _{n}^{2}} denotes R n {\displaystyle \mathbb {R} ^{n}} with the Euclidean norm (see the article on L p {\displaystyle L^{p}} spaces).

From this it follows that d ( X , Y ) n {\displaystyle d(X,Y)\leq n} for all X , Y Q ( n ) . {\displaystyle X,Y\in Q(n).} However, for the classical spaces, this upper bound for the diameter of Q ( n ) {\displaystyle Q(n)} is far from being approached. For example, the distance between n 1 {\displaystyle \ell _{n}^{1}} and n {\displaystyle \ell _{n}^{\infty }} is (only) of order n 1 / 2 {\displaystyle n^{1/2}} (up to a multiplicative constant independent from the dimension n {\displaystyle n} ).

A major achievement in the direction of estimating the diameter of Q ( n ) {\displaystyle Q(n)} is due to E. Gluskin, who proved in 1981 that the (multiplicative) diameter of the Banach–Mazur compactum is bounded below by c n , {\displaystyle c\,n,} for some universal c > 0. {\displaystyle c>0.}

Gluskin's method introduces a class of random symmetric polytopes P ( ω ) {\displaystyle P(\omega )} in R n , {\displaystyle \mathbb {R} ^{n},} and the normed spaces X ( ω ) {\displaystyle X(\omega )} having P ( ω ) {\displaystyle P(\omega )} as unit ball (the vector space is R n {\displaystyle \mathbb {R} ^{n}} and the norm is the gauge of P ( ω ) {\displaystyle P(\omega )} ). The proof consists in showing that the required estimate is true with large probability for two independent copies of the normed space X ( ω ) . {\displaystyle X(\omega ).}

Q ( 2 ) {\displaystyle Q(2)} is an absolute extensor. On the other hand, Q ( 2 ) {\displaystyle Q(2)} is not homeomorphic to a Hilbert cube.

See also

Notes

  1. Cube
  2. "The Banach–Mazur compactum is not homeomorphic to the Hilbert cube" (PDF). www.iop.org.

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