Definition: Given two Banach spaces X, and Y, the Banach Mazur distance is defined as where the infimum is taken over isomorphisms u from X to Y.

Remark: For finite dimensional spaces X, and Y of the same dimension the distance is finite and obtained. This can be proved using a) the fact that and b) using a compactness argument in the space L( ).

Examples: Let or . Then If p, and q are on different sides of 2, then the distance is achieved with the help of the discrete Fourier transform. This is a little bit easier in the complex case, for example:

Theorem: Let X be a finite dimensional space and d>0. Then there exists a subspace Y such that Proof:

Theorem (Fritz John). Let X be a n-dimensional Baach space. Then there exists a map u: l X such that ||u|| In fact, using the theory of 2-summing maps, one can show that u =RST, where R: X, T:X L are contractions and S: L L is the identity map for some probability measure.

Theorem: Let X be an n dimensional Banach space. Then there exists vectors x_1,..,x_n in the unit ball of X and x_1*,...,x_n* in the unit ball of X* such that In particular,

Corollary: Let X,Z be a finite dimensional Banach spaces of Y such that X has dimension n and for every element x_j in the Auerbach basis there exists a z_j in Z such that d(x_j,z_j)< Then X is isomorphic to a subspace of Z. Moreover, there exists an operator T from Y to Y such that T(X) Z ||T|| ||T || Proof:

Theorem: Let K be a compact topological space. Then there exists a net of linear maps u_t=v_tw_t on C(K) such that i) lim_t u_t(f)=f; ii) w_t:C(K) l is a positivity preserving contraction; iii) lim_t ||v_t||=1. Proof:

Corollary: Let X,Z be a finite dimensional Banach spaces of Y such that X has dimension n and for every element x_j in the Auerbach basis there exists a z_j in Z such that d(x_j,z_j)< Then X is isomorphic to a subspace of Z. Moreover, there exists an operator T from Y to Y such that T(X) Z ||T|| ||T || Proof:

Remark: It is not required to assume K metrizable, as long as we have Urysohn's Lemma. The proof this lemma is based on a topological requirement, namely that two disjoint compact set can be separated by two open sets. This holds in compact Haussdorf spaces.