normed vector space

**mms** · December 14th 2009, 01:22 PM

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

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

(iii) $ 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!

**Jose27** · December 14th 2009, 04:21 PM

Originally Posted by mms

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

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

(iii) $ 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 $n\in \mathbb{N}$ 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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