Asymptotic Theory Of Finite Dimensional Normed Spaces: Isoperimetric Inequalities In Riemannian Manifolds (lecture Notes In Mathematics)
by Vitali D. Milman /
2002 / English / PDF
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This book deals with the geometrical structure of finite
dimensional normed spaces, as the dimension grows to infinity. This
is a part of what came to be known as the Local Theory of Banach
Spaces (this name was derived from the fact that in its first
stages, this theory dealt mainly with relating the structure of
infinite dimensional Banach spaces to the structure of their
lattice of finite dimensional subspaces). Our purpose in this book
is to introduce the reader to some of the results, problems, and
mainly methods developed in the Local Theory, in the last few
years. This by no means is a complete survey of this wide area.
Some of the main topics we do not discuss here are mentioned in the
Notes and Remarks section. Several books appeared recently or are
going to appear shortly, which cover much of the material not
covered in this book. Among these are Pisier's [Pis6] where
factorization theorems related to Grothendieck's theorem are
extensively discussed, and Tomczak-Jaegermann's [T-Jl] where
operator ideals and distances between finite dimensional normed
spaces are studied in detail. Another related book is Pietch's
[Pie].
This book deals with the geometrical structure of finite
dimensional normed spaces, as the dimension grows to infinity. This
is a part of what came to be known as the Local Theory of Banach
Spaces (this name was derived from the fact that in its first
stages, this theory dealt mainly with relating the structure of
infinite dimensional Banach spaces to the structure of their
lattice of finite dimensional subspaces). Our purpose in this book
is to introduce the reader to some of the results, problems, and
mainly methods developed in the Local Theory, in the last few
years. This by no means is a complete survey of this wide area.
Some of the main topics we do not discuss here are mentioned in the
Notes and Remarks section. Several books appeared recently or are
going to appear shortly, which cover much of the material not
covered in this book. Among these are Pisier's [Pis6] where
factorization theorems related to Grothendieck's theorem are
extensively discussed, and Tomczak-Jaegermann's [T-Jl] where
operator ideals and distances between finite dimensional normed
spaces are studied in detail. Another related book is Pietch's
[Pie].