I think I have to take two points in and show that any mapping from to would result in a different distance between them.Quote:
Let with , but all other distances equal to 1.
d is a metric.
Prove that the metric space X is not isometric to any subset of for any n.
Hint: can you realise as a subset of a sphere of appropriate radius, with the spherical "great circle" metric?
Here's my attempt:
Take a sphere of radius 1 and the points A and D.
Since , A and D are on opposite sides of the sphere.
Suppose there exists an isometry where is is the plane used in stereographic projection (also a subset of ).
Then . Hence which is not equal to .
Hence cannot be an isometry.