1. ## 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!

2. Originally Posted by mms
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 $\displaystyle k\neq l$)