# Isometric imbedding

• Nov 16th 2009, 02:21 AM
math.dj
Isometric imbedding
Let X,Y be metric spaces with metrics d1,d2 respectively.
let f:X-->Y be a function with the property that for all pair of points x1,x2 of X, d2(f(x1),f(x2))=d1(x1,x2).
Show that f is an imbedding..

• Nov 16th 2009, 03:00 AM
Sampras
Quote:

Originally Posted by math.dj
Let X,Y be metric spaces with metrics d1,d2 respectively.
let f:X-->Y be a function with the property that for all pair of points x1,x2 of X, d2(f(x1),f(x2))=d1(x1,x2).
Show that f is an imbedding..

So $Ld_{1}(x,y) \leq d_{2}(f(x_1), f(x_2)) \leq Cd_{1}(x,y)$ which is definition of embedding.