Finite Dimensional Normed Spaces, Preliminaries

Abstract

Let X, Y be two n-dimensional normed spaces. The Banach-Mazur distance between them is defined as

$$ d(X,Y) = inf\{ \parallel T\parallel \cdot \parallel T^{ - 1} \parallel ; T:X \to Y isomorphism\} . $$

Obviously d(X, Y) ≥ 1 and d(X, Y) = 1 if and only if X and Y are isometric. If d(X, Y) ≤ λ we say that X and Y are λ — isomorphic. The notion of the distance also has a geometrical interpretation. If d(X, Y) is small then in some sense the two unit balls B(X) = {x ∈ X; ‖x‖ ≤ 1} and B(Y) = {y ∈ Y; ‖y‖ ≤ 1} are close one to the other. More precisely there is a linear transformation ϕ such that

$$ B(X) \subseteq \varphi (B(Y)) \subseteq d(X,Y)B(X). $$