Results 1 to 7 of 7

Math Help - [SOLVED] Metric Space basic definition

  1. #1
    Member
    Joined
    Jul 2008
    Posts
    119

    [SOLVED] Metric Space basic definition

    Problem:
    Let  X = C([0,1 ], R) . Find d(f,g) for each of the following pairs of functions:
    a.  f(x) =x and g(x)=cos(x) for each x in [0, 1]
    b. I did not bother to write down this part; I just need to understand the notation properly.
    =======================
    Definition:

     d(p,q) = \sqrt{\sum_{i=1}^n (p_{i}-q{i} )^2}

    So for part (a), I let  p_{i} = x and  q_{i}=cosx and n = 1, since this is in \mathbb{R}.

    d(f, g) = \sqrt{\sum_{i=1}^1 (x - cos(x))^2}

    but here I am confused. Would the d(f, g) be

    d(f, g) = \sqrt{(1-cos(1))^2 + (0 - cos(0))^2} ?

    Thank you for your help. Much appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1581
    Awards
    1
    Quote Originally Posted by Paperwings View Post
    Problem: Let  X = C([0,1 ], R) . Find d(f,g) for each of the following pairs of functions:  f(x) =x and g(x)=cos(x) for each x in [0, 1]
    I think that you have not understood the definition of the metric on this space.
    Do you understand the there are many different ways to define metrics?
    One sees the so called supremum metric for this space of continuous functions:
    d\left( {f,g} \right) = \mathop {\sup }\limits_{x \in \left[ {0,1} \right]} \left| {f(x) - g(x)} \right|.

    Here is another possible metric: d\left( {f,g} \right) = \int\limits_0^1 {\left| {f(x) - g(x)} \right|dx} .

    Thus you need to see what is the definition for the metric your textbook expects you to use on the space.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jul 2008
    Posts
    119
    For  C([a,b],R) , my textbook defines the metric space as

    d(f,g) = max { \left| f(x)-g(x) \right| | x \in [a,b]}, which is the supremum metric that you've pointed out.

    So, pertaining to the problem, d(f,g) it is asking for the greatest/maximum distance between the function x and cosx in the interval [0, 1], correct?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1581
    Awards
    1
    Quote Originally Posted by Paperwings View Post
    For  C([a,b],R) So, pertaining to the problem, d(f,g) it is asking for the greatest/maximum distance between the function x and cosx in the interval [0, 1], correct?
    Yes that is correct.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Jul 2008
    Posts
    119
    Plato, is there a way to find the maximum distance of two functions mathematically instead of looking at the graphs such as a formula? For in this problem, I can tell by graphing that d(f, g) = 1 for f(x) = x and g(x) = cosx since at x = 0, then f(0) = 0, and f(x) = 1.

    If for example if f(x) = 4x^4 and  g(x) = 6x^2 -3x for  x \in [0, 1] , how would I find the maximum distance? Thank you.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1581
    Awards
    1
    Well, you have done that is basic calculus. Have you not?
    Find the max of f(x)-g(x) and g(x)-f(x) to account for the absolute value.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Jul 2008
    Posts
    119
    I see. At first, thinking of an integral formula but that made no sense. Thank you.
    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. 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
  3. Sets > Metric Space > Euclidean Space
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 25th 2010, 10:17 PM
  4. [SOLVED] Metric Space- Distance Function
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: December 15th 2009, 08:32 PM
  5. [SOLVED] Metric Space and Inequality
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: July 31st 2008, 04:08 PM

Search Tags


/mathhelpforum @mathhelpforum