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Math Help - normed vector space

  1. #1
    mms
    mms is offline
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    normed vector space

    show that in the vector space <br />
\ell _1  = \left\{ {r_k \,\,:\,\sum {\left| {r_k } \right|} \,converges} \right\}<br />

    (i) <br />
\left\| {r_k } \right\| = \sum {r_k } <br />
is a norm
    (ii) <br />
P = \left\{ {\left\{ {x_k } \right\}\,:\,x_k  \geqslant 0\,\forall k \in \mathbb{N}} \right\}<br />
has en empty interior

    (iii) <br />
B\left( {\vec 0,1} \right)<br />
is not compact

    please help im trying to learn this by myself and i would like help on how to do this problems

    thanks!
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  2. #2
    Super Member
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    Quote Originally Posted by mms View Post
    show that in the vector space <br />
\ell _1  = \left\{ {r_k \,\,:\,\sum {\left| {r_k } \right|} \,converges} \right\}<br />

    (i) <br />
\left\| {r_k } \right\| = \sum {r_k } <br />
is a norm
    (ii) <br />
P = \left\{ {\left\{ {x_k } \right\}\,:\,x_k  \geqslant 0\,\forall k \in \mathbb{N}} \right\}<br />
has en empty interior

    (iii) <br />
B\left( {\vec 0,1} \right)<br />
is not compact

    please help im trying to learn this by myself and i would like help on how to do this problems

    thanks!
    For (i) just remmeber that for all n\in \mathbb{N} we have \vert x_n + y_n \vert \leq \vert x_n\vert + \vert y_n \vert so adding we get \sum_{i=1}^{n} \vert x_i+y_i \vert \leq \sum_{i=1}^{n} \vert x_i \vert + \vert y_i \vert \leq \sum_{i=1}^{\infty } \vert x_i \vert + \vert y_i \vert < \infty

    For (ii) take a sequence y^k=(y_{n}^k) \in \ell _1 such that y_{n}^k = x_n if n\neq k and y_{n}^k= -\frac{1}{k} if n=k then \Vert x-y^k \Vert = \vert x_k + \frac{1}{k} \vert \rightarrow 0 as k\rightarrow \infty (since x_n \rightarrow 0 )

    For (iii) what can you say about the sequence x^k=(x_{n}^k) where x_{n}^k=0 if n\neq k and x_{n}^k=\frac{1}{2} if n=k (Notice that \Vert x^k - x^l \Vert = 1 for all k\neq l)
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