Gon alo rodrig2007-12-19 17:53:49

Hi all,

Are there any results on the classification of Banach spaces within

isometric isomorphism? For finite dimensional spaces (fixing the

dimension) the l^p spaces, for p\in [1, \infty] are pairwise

non-isometric isomorphic. Are there any results on the isometric

classification for finite dimensional Banach spaces?

With my best regards,

G. Rodrigues

Robert israel2007-12-19 17:54:16

If U is any bounded convex balanced (i.e. -U = U) open set, the Minkowski

gauge p_U(x) = inf{s: x in sU} is a norm, and conversely any norm is

the Minkowski gauge for its unit ball. So the isometric classification

for finite-dimensional Banach spaces is equivalent to the classification

of bounded convex balanced open sets under linear transformations.

I think there is too much freedom here to have any really useful

parametrization.

Robert Israel israel@math.ubc.ca

Department of Mathematics http://www.math.ubc.ca/~israel

University of British Columbia

Vancouver, BC, Canada V6T 1Z2

Gon alo rodrig2007-12-20 06:50:19

Then let me vary the question slightly. Has anybody put any sensible

topological structure on the “moduli space” of bounded balanced convex

sets modulo linear transformations?

With my best regards,

G. Rodrigues

Stephen montgo2007-12-20 17:50:46

Banach spaces are usually studied from two points of view: the infinite

dimensional theory, which usually considers Banach spaces up to isomorphism, and

the local theory which considers finite dimensional spaces, and considers

properties that hold up to a constant as the dimension tends to infinity. Both

of these are very rich and difficult subjects. The infinite dimensional theory

in particular is fraught with many interesting and difficult counterexamples to

almost every reasonable conjecture, and the local theory is not much better.

One realises that there are some very strange Banach spaces out there, that make

the l^p spaces seem very simple. As such, any classification of Banach spaces

up to isometry is going to impossible at this present time. The isometric

theory of finite dimensional Banach spaces tends to be somewhat specialised, and

might, for example, ask questions like when finite dimensional l^p spaces are

subspaces of l^q or L^q for some 0

Dan luecking2007-12-20 17:51:27

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The Banach-Mazur distance: Take f(X,Y) = \inf \norm{L}\norm{L^{-1}} over all isomorphisms L between the spaces X and Y. If the unit ball of one is a linear image of the unit ball of the other, this infimum is 1 otherwise it is greater than 1 (in finite dimension). So \log f(X,Y) would seem to be a reasonable measure of distance. I don't know if it is indeed a metric, but it shouldn't be hard to check. Dan -- Dan Luecking Department of Mathematical Sciences University of Arkansas Fayetteville, Arkansas 72701 luecking at uark dot edu

The Banach-Mazur distance: Take f(X,Y) = \inf \norm{L}\norm{L^{-1}} over all isomorphisms L between the spaces X and Y. If the unit ball of one is a linear image of the unit ball of the other, this infimum is 1 otherwise it is greater than 1 (in finite dimension). So \log f(X,Y) would seem to be a reasonable measure of distance. I don't know if it is indeed a metric, but it shouldn't be hard to check. Dan -- Dan Luecking Department of Mathematical Sciences University of Arkansas Fayetteville, Arkansas 72701 luecking at uark dot edu

Gowan42007-12-21 21:12:23

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A few non-trivial topological propterties of this compact metric space are known. In the paper "The TOpology of the Banach-Mazur Compactum" by S. A. Antonyan, _Fundamenta Mathematicae_ vol. 166, (2000), pp. 209 - 232, it is proved that the Banach-Mazur compactum (Minkowski compactum) of two-dimensional Banach spaces is a non-homogeneous absolute retract. This contradicts a folk belief that it would be homeomorphic to the Hilbert cube.

A few non-trivial topological propterties of this compact metric space are known. In the paper "The TOpology of the Banach-Mazur Compactum" by S. A. Antonyan, _Fundamenta Mathematicae_ vol. 166, (2000), pp. 209 - 232, it is proved that the Banach-Mazur compactum (Minkowski compactum) of two-dimensional Banach spaces is a non-homogeneous absolute retract. This contradicts a folk belief that it would be homeomorphic to the Hilbert cube.

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