Let be a metric space, prove that is a metric over Moreover is bounded with the metric and

First part is easy, now I wanted to see how to prove that is bounded with the metric is it because ? Now since so since then then is this correct?

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- Mar 29th 2011, 01:42 PMConnectedMetric space
Let be a metric space, prove that is a metric over Moreover is bounded with the metric and

First part is easy, now I wanted to see how to prove that is bounded with the metric is it because ? Now since so since then then is this correct? - Mar 29th 2011, 01:49 PMDrexel28
- Mar 29th 2011, 02:11 PMConnected
I actually edited that before, probably you didn't see it, so it was okay anyway!

- Mar 29th 2011, 08:18 PMbkarpuz
In general, you can prove this:

Let be a metric space, and be a increasing positive concave function with , roughly,

- on
- on .

Then is a metric space, where defined by for . (Itwasntme)

The first condition is required for M1, and the last two is for M3,

you can see that M2 follows directly without any condition. :)

This was what I have proved when I was 2nd year BC. :)

You can also show what if . :)