Results 1 to 2 of 2

Thread: normed vector space

  1. #1
    mms
    mms is offline
    Junior Member
    Joined
    Jul 2009
    Posts
    67

    normed vector space

    show that in the vector space $\displaystyle
    \ell _1 = \left\{ {r_k \,\,:\,\sum {\left| {r_k } \right|} \,converges} \right\}
    $

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

    (iii)$\displaystyle
    B\left( {\vec 0,1} \right)
    $ is not compact

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

    thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Quote Originally Posted by mms View Post
    show that in the vector space $\displaystyle
    \ell _1 = \left\{ {r_k \,\,:\,\sum {\left| {r_k } \right|} \,converges} \right\}
    $

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

    (iii)$\displaystyle
    B\left( {\vec 0,1} \right)
    $ 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 $\displaystyle n\in \mathbb{N}$ we have $\displaystyle \vert x_n + y_n \vert \leq \vert x_n\vert + \vert y_n \vert$ so adding we get $\displaystyle \sum_{i=1}^{n} \vert x_i+y_i \vert \leq \sum_{i=1}^{n} \vert x_i \vert + \vert y_i \vert \leq \sum_{i=1}^{\infty } \vert x_i \vert + \vert y_i \vert < \infty$

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

    For (iii) what can you say about the sequence $\displaystyle x^k=(x_{n}^k)$ where $\displaystyle x_{n}^k=0$ if $\displaystyle n\neq k$ and $\displaystyle x_{n}^k=\frac{1}{2}$ if $\displaystyle n=k$ (Notice that $\displaystyle \Vert x^k - x^l \Vert = 1$ for all $\displaystyle k\neq l$)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. A question on normed vector space
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Jul 22nd 2011, 12:19 PM
  2. Normed space and dual space
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: Jun 5th 2011, 10:46 PM
  3. Problem with showing a normed vector space is complete
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Nov 25th 2009, 01:36 PM
  4. Subspace of Normed Vector Space
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: Sep 10th 2009, 01:57 PM
  5. Normed space
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: Mar 14th 2009, 03:52 PM

Search Tags


/mathhelpforum @mathhelpforum