text: 1. ## normed vector space
  
  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!
  
  2. 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 $k\neq l$)
explanation: The document contains a series of mathematical problems related to normed vector spaces, specifically the space \(\ell_1\). The problems are:
  
  1. Show that \(\left\| {r_k } \right\| = \sum {r_k }\) is a norm.
  2. Show that \(P = \left\{ {\left\{ {x_k } \right\}\,:\,x_k \geqslant 0\,\forall k \in \mathbb{N}} \right\}\) has an empty interior.
  3. Show that \(B\left( {\vec 0,1} \right)\) is not compact.
  
  The document also includes answers to these questions, providing detailed steps and reasoning for each part.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 1. ## normed vector space
  
  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!
  
  2. 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 $k\neq l$)
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a series of mathematical problems related to normed vector spaces, specifically the space \(\ell_1\). The problems are:
  
  1. Show that \(\left\| {r_k } \right\| = \sum {r_k }\) is a norm.
  2. Show that \(P = \left\{ {\left\{ {x_k } \right\}\,:\,x_k \geqslant 0\,\forall k \in \mathbb{N}} \right\}\) has an empty interior.
  3. Show that \(B\left( {\vec 0,1} \right)\) is not compact.
  
  The document also includes answers to these questions, providing detailed steps and reasoning for each part.
  
  Label: multiple-question-and-answer