Results 1 to 4 of 4

Math Help - Norm on a Cartesian product

  1. #1
    Junior Member
    Joined
    Nov 2009
    From
    Pocatello, ID
    Posts
    59

    Norm on a Cartesian product

    Let V_1, V_2 be normed vector spaces with norms ||\cdot||_1 and ||\cdot||_2, respectively. Show that (||v_1||_1^p + ||v_2||_2^p)^\frac{1}{p}, 1\leq p < \infty defines a norm on the space V_1\times V_2.

    I've got the first three criteria down (positive definite, zero only when the vector is the zero vector, absolute homogeneity), however I'm having trouble seeing how to break this apart to show it satisfies the triangle inequality.

    Suggestions on how to get started?
    Last edited by mathematicalbagpiper; August 28th 2010 at 09:07 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by mathematicalbagpiper View Post
    Let V_1, V_2 be normed vector spaces with norms ||\cdot||_1 and ||\cdot||_2, respectively. Show that (||v_1||_1^p + ||v_2||_2^p)^\frac{1}{p}, 1\leq p < \infty defines a norm on the space V_1\times V_2.

    I've got the first three criteria down (positive definite, zero only when the vector is the zero vector, absolute homogeneity), however I'm having trouble seeing how to break this apart to show it satisfies the triangle inequality.
    Given two elements (u_1,u_2),\ (v_1,v_2) in V_1\times V_2, you need to show that the triangle inequality \bigl(\|u_1+v_1\|_1^p + \|u_2+v_2\|_2^p\bigr)^{1/p} \leqslant \bigl(\|u_1\|_1^p + \|u_2\|_2^p\bigr)^{1/p} + \bigl(\|v_1\|_1^p + \|v_2\|_2^p\bigr)^{1/p} holds.

    Use the triangle inequality in V_1 and V_2 to see that the left side of that inequality is \leqslant \bigl((\|u_1\|_1+\|v_1\|_1)^p + (\|u_2\|_2+\|v_2\|_2)^p\bigr)^{1/p}. Then use Minkowski's inequality in two-dimensional space to conclude that that is less than or equal to the right side of the triangle inequality.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2009
    From
    Pocatello, ID
    Posts
    59
    That does work out nicely, except for the fact that if I use that I won't get credit for it, since this exercise is in section 1.2 of the book and Minkowski's Inequality doesn't get introduced until Section 1.5, thus I don't have license to use it.

    (My professor is very picky about that).
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by mathematicalbagpiper View Post
    That does work out nicely, except for the fact that if I use that I won't get credit for it, since this exercise is in section 1.2 of the book and Minkowski's Inequality doesn't get introduced until Section 1.5, thus I don't have license to use it.

    (My professor is very picky about that).
    The general form of Minkowski's inequality may not come until Section 1.5, but are you sure that some special cases of it haven't come earlier? Your problem only needs the inequality for a 2-dimensional space. In fact, if V_1 and V_2 are both 1-dimensional spaces then the problem is actually equivalent to the 2-dim. Minkowski inequality. So somehow or other you will have to make use of it!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Inner Product space not induced by L1 norm
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 23rd 2010, 01:28 AM
  2. Norm versus inner product
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 17th 2010, 09:43 AM
  3. Infinity-norm and inner-product
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: February 28th 2010, 02:54 PM
  4. Proof (vectors, norm, and dot product)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 12th 2009, 09:58 AM
  5. Scalar Product/Norm proof
    Posted in the Calculus Forum
    Replies: 2
    Last Post: July 14th 2008, 06:25 PM

Search Tags


/mathhelpforum @mathhelpforum