1 19th December 19:53 gonçalo rodrigues External User   Posts: 1 Classification of Banach spaces 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

 2 19th December 19:54 robert israel External User   Posts: 1 Classification of Banach spaces 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
 3 20th December 08:50 gonçalo rodrigues External User   Posts: 1 Classification of Banach spaces 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
 4 20th December 19:50 stephen montgomery-smith External User   Posts: 1 Classification of Banach spaces 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
 5 20th December 19:51 dan luecking External User   Posts: 1 Classification of Banach spaces 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
 6 21st December 23:12 gowan4 External User   Posts: 1 Classification of Banach spaces 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.