Results 1 to 2 of 2

Math Help - Verify a function is a metric...

  1. #1
    Member
    Joined
    Nov 2008
    Posts
    121

    Verify a function is a metric...

    Let (X,\| \cdot \|) be a normed space.

    Show the function d(x,y)= \| x - y \| is a metric.
    Last edited by bigdoggy; September 25th 2009 at 02:16 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    Posts
    461
    Hi biddoggy!

    Quote Originally Posted by bigdoggy View Post
    Let (X<\| \cdot \|) be a normed space.

    Show the function d(x,y)= \| x - y \| is a metric.
    a metric satisfies:

    (1) d(x,x) = 0
    and d(x,y) \not= 0, if x \not= y

    So: ||x-x|| = ||0|| = 0
    OKAY

    ||x-y|| > 0, because  x \not= y, thus x-y \not= 0
    OKAY

    (2) d(x,y) = d(y,x)

    ||x-y|| =||y-x||

    ||x-y|| = ||-(x-y)||

    ||x-y|| = |-1|*||x-y|

    ||x-y|| = ||x-y||

    OKAY

    (3) d(x+z,y+z) \le d(x,y) + d(y,z)

    ||x+z - (y+z)|| \le ||x+z|| + ||-(y+z)|| = ||x+z|| + ||y+z||
    OKAY

    Are there any questions on that?

    Yours
    Rapha
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Verify that each function has given roots
    Posted in the Algebra Forum
    Replies: 2
    Last Post: October 20th 2010, 03:53 AM
  2. Limit of function from one metric space to another metric space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 17th 2010, 03:04 PM
  3. I need to verify part of the proof involve Green's function.
    Posted in the Advanced Applied Math Forum
    Replies: 0
    Last Post: September 12th 2010, 04:58 PM
  4. Replies: 2
    Last Post: September 2nd 2010, 11:28 AM
  5. Show that function is a pre-metric
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: April 1st 2010, 04:27 PM

Search Tags


/mathhelpforum @mathhelpforum