Results 1 to 5 of 5

Math Help - Is this a Metric space?

  1. #1
    pkr
    pkr is offline
    Junior Member
    Joined
    Nov 2008
    Posts
    49

    Is this a Metric space?

    X = R^2 d(x,y) = (x1 - y1)^2 + (x2 - y2)^2

    Having trouble proving the 3 axioms...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    May 2006
    Posts
    244
    Quote Originally Posted by pkr View Post
    X = R^2 d(x,y) = (x1 - y1)^2 + (x2 - y2)^2

    Having trouble proving the 3 axioms...
    The first two axioms are trivial, so you are left to prove:

    d(x,y)+d(y,z) \ge d(x,z)

    but d(a,b)=[\rho(a,b)]^2 where \rho is the usual metric on \mathbb{R}^2

    So:

    d(x,y)+d(y,z) = [\rho(x,y)]^2+[\rho(y,z)]^2=[\rho(x,y)+\rho(y,z)]^2-2\rho(x,y)\rho(y,z)  \ge [\rho(x,y)+\rho(y,z)]^2

    ....................... \ge [\rho(x,z)]^2=d(x,z)

    .
    Follow Math Help Forum on Facebook and Google+

  3. #3
    pkr
    pkr is offline
    Junior Member
    Joined
    Nov 2008
    Posts
    49
    Thanks for that but what exactly is p? I've never seen that notation before, is it the same as sqrt(a^2 + b^2)?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    May 2006
    Posts
    244
    Quote Originally Posted by pkr View Post
    Thanks for that but what exactly is p? I've never seen that notation before, is it the same as sqrt(a^2 + b^2)?
    It is what I said, the normal metric on \mathbb{R}^2:

    \rho(a,b)=\sqrt{(a_1-b_1)^2+(a_2-b_2)^2}

    .
    Follow Math Help Forum on Facebook and Google+

  5. #5
    pkr
    pkr is offline
    Junior Member
    Joined
    Nov 2008
    Posts
    49
    Ah, sorry! Thanks very much.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: July 8th 2011, 02:16 PM
  2. Is it a metric space?
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: April 3rd 2011, 06:44 PM
  3. Limit of function from one metric space to another metric space
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: September 17th 2010, 02:04 PM
  4. Sets > Metric Space > Euclidean Space
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 25th 2010, 10:17 PM
  5. Metric Space
    Posted in the Calculus Forum
    Replies: 5
    Last Post: January 11th 2009, 03:47 AM

Search Tags


/mathhelpforum @mathhelpforum