Editing Embedding (section)

==Metric spaces==

A mapping <math>\phi: X \to Y</math> of [[metric spaces]] is called an ''embedding''
(with [[stretch factor|distortion]] <math>C>0</math>) if
:<math> L d_X(x, y) \leq d_Y(\phi(x), \phi(y)) \leq CLd_X(x,y) </math>
for every <math>x,y\in X</math> and some constant <math>L>0</math>.

=== Normed spaces ===

An important special case is that of [[normed spaces]]; in this case it is natural to consider linear embeddings.

One of the basic questions that can be asked about a finite-dimensional [[normed space]] <math>(X, \| \cdot \|)</math> is, ''what is the maximal dimension <math>k</math> such that the [[Hilbert space]] <math>\ell_2^k</math> can be linearly embedded into <math>X</math> with constant distortion?''

The answer is given by [[Dvoretzky's theorem]].