gonçalo rodrigues

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 israel

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

gonçalo rodrigues

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 montgomery-smith

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<p,q<=infinity.

The local theory studies n-dimensional Banach spaces endowed with a metric that
says how non-isometric they are. This is often called the Minkowski compactum
(because the space is compact). Typical problems include finding the diameter
of this metric space, and the methods usually involve constructing n-dimensional
Banach spaces at random and showing that they have certain properties with
positive probability. If you want to get a flavor of this subject, you could
try looking at the following book.

Milman, Vitali D.(IL-TLAV); Schechtman, Gideon(IL-WEIZ)
Asymptotic theory of finite-dimensional normed spaces.
With an appendix by M. Gromov. Lecture Notes in Mathematics, 1200.
Springer-Verlag, Berlin, 1986. viii+156 pp. \$15.00. ISBN 3-540-16769-2

Gilles Pisier in reviewing this book in Math Reviews concludes by saying: In
conclusion, this is an excellent book, which admirably achieves its goal of
introducing the reader to the diverse methods of local theory. Although many
questions remain open, this publication marks a first stage of maturity for this
theory. It is clearly a "must" for specialists, but it deserves a large
diffusion among other mathematicians, and it can serve as the basis for an
advanced graduate course.

dan luecking

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

gowan4

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.