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Math Help - Bounded spaces

  1. #1
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    Bounded spaces

    I have proved that a totally bounded metric space is bounded using the idea of epsilon nets, but can't work the attached out.

    Any solutions would be great thanks.
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  2. #2
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    Quote Originally Posted by Cairo View Post
    I have proved that a totally bounded metric space is bounded using the idea of epsilon nets, but can't work the attached out.

    Any solutions would be great thanks.
    Notice that d(e_k,e_l)=\sqrt{2} if k\neq l now try to find a \epsilon-net with \epsilon < \sqrt{2}. That it's bounded should be trivial.
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  3. #3
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    Where did you get root(2) from? I thought d(a,b) was root(1)????
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  4. #4
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    Quote Originally Posted by Cairo View Post
    Where did you get root(2) from? I thought d(a,b) was root(1)????
    If k\neq l then d(e_k,e_l)= \left( \sum_{n=1}^{\infty } |e_{k}^n - e_{l}^n|^2 \right) ^{\frac{1}{2} } = (1+1)^{\frac{1}{2} } because e_k and e_l have their respective 1 in different places.

    Also it doens't matter which number it is, just that each point is a constant distance away from each other, and so any net we try to build with a number smaller than this constant will fail to cover all from a finite point set.
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  5. #5
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    Okay....but i'm still not sure how the proof would go.

    Sorry for being thick here!
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  6. #6
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    Quote Originally Posted by Cairo View Post
    Okay....but i'm still not sure how the proof would go.

    Sorry for being thick here!
    Assume it is totally bounded and pick a number 0<a<\sqrt{2} then we must have that there exist e_{m_1},...,e_{m_k} \in (e_j)_{j\in \mathbb{N} } such that (e_j)_{j\in \mathbb{N} } \subseteq \cup_{i=1}^{k} B_a(e_{m_i} )  but B_a(e_{m_i} ) = \{ y \in l_2 : d(y,e_{m_i})<a \} and so (e_j)_{j\in \mathbb{N} } \cap B_a(e_{m_i} ) = \{ e_{m_i} \}
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