# Isometric imbedding

• November 16th 2009, 03:21 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..

thnak you in advance
• November 16th 2009, 04:00 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..

thnak you in advance

Take L = C = 1.

So $Ld_{1}(x,y) \leq d_{2}(f(x_1), f(x_2)) \leq Cd_{1}(x,y)$ which is definition of embedding.
• November 16th 2009, 04:09 AM
math.dj
thank you for ur response..but we have not yet come across this definition of embedding..
• November 18th 2009, 02:18 AM
math.dj
i need a little help..i have shown that f is injective and continuous.now i need to show that g:X-->f(x) is a homeomorphism..how do i show g and g^-1(that is g inverse) are continuous..