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Math Help - 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 l_\infty(R) . If they are, verify by axioms. If not, provide counter example.


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

    (i) || ||_* defined by ||x||_*=|x_1|

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

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

     y \in l_\infty(R)

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

    ||x+y||_*=||(x_n+y_n)||_*=|x_n+y_n|


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

     |x_n+y_n|\le|x_n|+|y_n|

     \le k_x+k_y \forall n \in N

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


    b) ||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 l_\infty(R) . If they are, verify by axioms. If not, provide counter example.


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

    (i) || ||_* defined by ||x||_*=|x_1|

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

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

     y \in l_\infty(R)

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

    ||x+y||_*=||(x_n+y_n)||_*=|x_n+y_n|


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

     |x_n+y_n|\le|x_n|+|y_n|

     \le k_x+k_y \forall n \in N

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


    b) ||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, \|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, \|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

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

     \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.
    ||x||_*=|x_1| \ge 0 iff |x_1| \ge 0
     \forall (x_n) \in R..?
    That is what confused me. Usually we just use three properties for norm.

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

    Then \|t\|_*=|0|=0...?
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  6. #6
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    Re: Vector Space of Bounded Sequences

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