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Thread: Vector Space of Bounded Sequences

  1. #1
    Senior Member bugatti79's Avatar
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    Vector Space of Bounded Sequences

    Folks,

    Consider the vector space l_infty(R) of all bounded sequences. Decide whether or not the following norm is defined on $\displaystyle l_\infty(R)$ . If they are, verify by axioms. If not, provide counter example.


    $\displaystyle x \in l_{\infty} (R); x=(x_n)$,

    (i) $\displaystyle || ||_*$ defined by $\displaystyle ||x||_*=|x_1|$

    a) Since $\displaystyle x \in l_\infty(R)$

    $\displaystyle \exists k_x >0$ s.t $\displaystyle |x_n| \le k_x \forall n \in N$

    $\displaystyle y \in l_\infty(R)$

    $\displaystyle \exists k_y>0$ s.t $\displaystyle |y_n| \le k_y \forall n \in N$

    $\displaystyle ||x+y||_*=||(x_n+y_n)||_*=|x_n+y_n|$


    Now we have that $\displaystyle |a+b| \le |a|+|b|$, therefore

    $\displaystyle |x_n+y_n|\le|x_n|+|y_n|$

    $\displaystyle \le k_x+k_y \forall n \in N $

    $\displaystyle \le ||x||_*+||y||_*$


    b) $\displaystyle ||ax||_*=|ax_1|=|a| |x_1|=|a |||x||_*$

    Axioms c) and d) I dont know how to attempt for this space...?
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  2. #2
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    Re: Vector Space of Bounded Sequences

    Quote Originally Posted by bugatti79 View Post



    Consider the vector space l_infty(R) of all bounded sequences. Decide whether or not the following norm is defined on $\displaystyle l_\infty(R)$ . If they are, verify by axioms. If not, provide counter example.


    $\displaystyle x \in l_{\infty} (R); x=(x_n)$,

    (i) $\displaystyle || ||_*$ defined by $\displaystyle ||x||_*=|x_1|$

    a) Since $\displaystyle x \in l_\infty(R)$

    $\displaystyle \exists k_x >0$ s.t $\displaystyle |x_n| \le k_x \forall n \in N$

    $\displaystyle y \in l_\infty(R)$

    $\displaystyle \exists k_y>0$ s.t $\displaystyle |y_n| \le k_y \forall n \in N$

    $\displaystyle ||x+y||_*=||(x_n+y_n)||_*=|x_n+y_n|$


    Now we have that $\displaystyle |a+b| \le |a|+|b|$, therefore

    $\displaystyle |x_n+y_n|\le|x_n|+|y_n|$

    $\displaystyle \le k_x+k_y \forall n \in N $

    $\displaystyle \le ||x||_*+||y||_*$


    b) $\displaystyle ||ax||_*=|ax_1|=|a| |x_1|=|a |||x||_*$

    Axioms c) and d) I dont know how to attempt for this space...?
    I am not sure what you mean by "Axioms c) and d)"

    But have you considered, $\displaystyle \|x\|_*=0\text{ if and only if }x=0~? $
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  3. #3
    Senior Member bugatti79's Avatar
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    Re: Vector Space of Bounded Sequences

    Quote Originally Posted by Plato View Post
    I am not sure what you mean by "Axioms c) and d)"

    But have you considered, $\displaystyle \|x\|_*=0\text{ if and only if }x=0~? $
    I shouldnt have labelled them c and d. I just meant the 4 general axioms of a norm.
    I think what you have wrote is one of them.

    is the final one

    $\displaystyle ||x||_*=|x_1| \ge 0$ iff $\displaystyle |x_1| \ge 0$

    $\displaystyle \forall (x_n) \in R$..?

    Thanks
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    Re: Vector Space of Bounded Sequences

    Quote Originally Posted by bugatti79 View Post
    I just meant the 4 general axioms of a norm.
    $\displaystyle ||x||_*=|x_1| \ge 0$ iff $\displaystyle |x_1| \ge 0$
    $\displaystyle \forall (x_n) \in R$..?
    That is what confused me. Usually we just use three properties for norm.

    Let $\displaystyle t=(0,1,1,\cdots,1\cdots)$ what is $\displaystyle \|t\|_*=~?$
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  5. #5
    Senior Member bugatti79's Avatar
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    Re: Vector Space of Bounded Sequences

    Quote Originally Posted by Plato View Post
    That is what confused me. Usually we just use three properties for norm.

    Let $\displaystyle t=(0,1,1,\cdots,1\cdots)$ what is $\displaystyle \|t\|_*=~?$
    Well since $\displaystyle ||||_*$ is defined by $\displaystyle ||x||_*=|x_1|$

    Then $\displaystyle \|t\|_*=|0|=0$...?
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    Re: Vector Space of Bounded Sequences

    Quote Originally Posted by bugatti79 View Post
    Well since $\displaystyle ||||_*$ is defined by $\displaystyle ||x||_*=|x_1|$
    Then $\displaystyle \|t\|_*=|0|=0$...?
    Well can $\displaystyle \bold{t}=\bold{0}~?$
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