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Metric Spaces (Proof)

  1. Mar 14, 2010 #1
    1. The problem statement, all variables and given/known data

    Let [tex](X_k,d_k)[/tex], [tex]1\leq k<\infty [/tex] be metric spaces
    Let [tex]X=\prod _{k=1}^{\infty} X_k [/tex] be their Cartesian product,
    that is, let [tex]X[/tex] be the set of sequences [tex](x_1,x_2,...)[/tex], where [tex]x_j\in X_j[/tex] for [tex]1\leq j < \infty [/tex]

    Show that a sequence [tex] \left\{x^{(k)}\right\}_{k=1}^{ \infty } [/tex] converges in [tex]X[/tex] if and only if [tex] \left\{x_j^{(k)}\right\}_{k=1}^{ \infty } [/tex] converges in [tex] X_j [/tex] for each [tex] j \geq 1. [/tex]


    2. Relevant equations



    3. The attempt at a solution

    Assume [tex] \left\{x^{(k)}\right\}_{k=1}^{ \infty } [/tex] converges in [tex]X[/tex].
    then [tex] \left\{x^{(k)}\right\}_{k=1}^{ \infty } = (a_{11},a_{21},...),(a_{12},a_{22},...),...[/tex] where [tex] a_{ik} \in X_i [/tex]
    So, [tex] \left\{x^{(k)}\right\}_{k=1}^{ \infty } [/tex] converges to [tex]c_i [/tex]

    Assume [tex] \left\{x_j^{(k)}\right\}_{k=1}^{ \infty } [/tex] converges in [tex]X_j [/tex] for each [tex] j \geq 1 [/tex]
    so, [tex] \left\{x^{(k)}\right\}_{k=1}^{ \infty } = (a_{11},a_{21},...),(a_{12},a_{22},...),...[/tex] where [tex] \left\{x_j^{(k)}\right\}_{k=1}^{ \infty } = a_{j1},a_{j2},...[/tex] converges to [tex]c_j[/tex]
    So, [tex] \left\{x^{(k)}\right\}_{k=1}^{ \infty } [/tex] converges to [tex] (c_1,c_2,...) [/tex]
     
  2. jcsd
  3. Mar 14, 2010 #2

    jbunniii

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    Can you tell us what is the metric on [itex]X[/itex]?
     
  4. Mar 14, 2010 #3
    I don't think it gives an explicit metric on X, but from part a (this is part b), it asks to show that something is a metric on X, but I didn't think that carried over (but maybe it does)

    from part a:
    Show that

    [tex] d(x,y) = \sum _{j=1}^{ \infty }\frac{1}{2^j}min(1,d_j(x_j,y_j)) [/tex]

    is a metric on [tex]X[/tex]

    (idk why it kept isametric in the tex tags :(
     
  5. Mar 14, 2010 #4

    jbunniii

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    I think it must carry over, because there's more than one way to define a metric on [itex]X[/itex] and it's hard to talk about convergence in a metric space if you don't specify what the metric is. Try proceeding under that assumption and let us know if you get stuck.
     
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