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Math Help - is this true for all norm on Rn

  1. #1
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    is this true for all norm on Rn

    Consider a norm on Rn.

    there are two vectors x,y in Rn
    x=(x1,x2,x3,...,xn)
    y=(y1,y2,...,yn)

    if, y1>x1,
    y2>x2,
    y3>x3,
    .
    .
    .
    yn>xn
    .
    is it true that norm : ||y||>= ||x|| for any norm on Rn?
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  2. #2
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    Quote Originally Posted by szpengchao View Post
    Consider a norm on Rn.

    there are two vectors x,y in Rn
    x=(x1,x2,x3,...,xn)
    y=(y1,y2,...,yn)

    if, y1>x1,
    y2>x2,
    y3>x3,
    .
    .
    .
    yn>xn
    .
    is it true that norm : ||y||>= ||x|| for any norm on Rn?
    consider \| y \|^2 how does it compare to \| x \|^2? what does that tell you?
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  3. #3
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    not defined

    ||y|| is not defined as \sqrt( y1^2+y2^2...+yn^2)

    so how can u compare ||y||,||x||
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    Quote Originally Posted by szpengchao View Post
    ||y|| is not defined as \sqrt( y1^2+y2^2...+yn^2)

    so how can u compare ||y||,||x||
    yes, i know what the norm is defined as. i asked you to compare \| y \|^2 with \| x \|^2 (we can do this since we were given comparisons component wise).

    anyway, just make a counter example, it shouldn't be hard. consider vectors in \mathbb{R}^2 given by

    \vec y = \left< 1,1 \right> and \vec x = \left< -5, -5 \right>

    do those vectors satisfy the conditions given? is \| y \| \ge \| x \| ?
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  5. #5
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    .

    i m sorry. forgot one condition:

    y1,y2,...,>0
    x1,x2,...,>0

    is there any counterexample ?
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    Quote Originally Posted by szpengchao View Post
    i m sorry. forgot one condition:

    y1,y2,...,>0
    x1,x2,...,>0

    is there any counterexample ?
    that's a big condition to forget

    in that case, note that y_i > x_i \implies y_i^2 > x_i^2 for 1 \le i \le n

    now go back and do what i told you to do before
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  7. #7
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    no idea

    no idea. i said the norm is not given as:

    \sqrt( y1^2+y2^2+...+yn^2)
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    Quote Originally Posted by szpengchao View Post
    no idea. i said the norm is not given as:

    \sqrt( y1^2+y2^2+...+yn^2)
    and if you square that, what do you get?
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  9. #9
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    i dont know

    i dont know
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  10. #10
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    Quote Originally Posted by szpengchao View Post
    i dont know
    you don't know what, say, (\sqrt{a})^2 is? what if i asked you to find (\sqrt{3})^2, you wouldn't be able to tell me?
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  11. #11
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    ...

    i know what you mean, but i m not talking about euclidean norm.
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    Quote Originally Posted by szpengchao View Post
    i know what you mean, but i m not talking about euclidean norm.
    we are dealing with the Euclidean n-space, why do you think this is not the Euclidean norm. and even if not, how does that matter. we have a definition with a square root in it and i asked you to square it. how does it matter what kind of definition it is?
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  13. #13
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    The question is

    The question is about a norm on Rn, but that norm needs not to be Euclidean. That argument is obviously true if it is euclidean norm.
    But there are other norms on Rn, i just want to find a counterexample.
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  14. #14
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    Quote Originally Posted by szpengchao View Post
    The question is about a norm on Rn, but that norm needs not to be Euclidean. That argument is obviously true if it is euclidean norm.
    But there are other norms on Rn, i just want to find a counterexample.
    you told me the definition of norm you are using. it is the default. we take that definition unless another one is specified
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    \begin{array}{ccl} \|y \|^2 & = &  \left( \sqrt{y_1^2 + y_2^2 + \cdots + y_n^2} \right)^2 \\<br />
& & \\<br />
& = & y_1^2 + y_2^2 + \cdots + y_n^2 \\<br />
& & \\<br />
& \ge & x_1^2 + x_2^2 + \cdots + x_n^2 \\ <br />
& & \\<br />
& = & \| x \|^2 \end{array}

    the result follows by taking the square roots of both sides
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