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Math Help - About the continuity of coordinate projection

  1. #1
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    About the continuity of coordinate projection

    I have attempted to solve this problem; however I couldn't do it.

    Please, could someone give me a solution or a partial solution?

    Thank you.

    Consider the vector space of all sequence of complex numbers. Let ||.|| be a norm in such space. Why at most a finite number of coordinate projections can be continuous (or b.ounded) in the norm ||.||?
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  2. #2
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    Without loss of generality assume that every projection is bounded with a constant K_i>0, which means that for all x=(x_1,x_2,...)
    || x_i e_i || <= K_i || x ||

    where e_i is the i-th unit vector.

    Then in particular for the vector x, defined by x_i:=i*K_i/|| e_i ||

    || x_i e_i ||=i*K_i <= K_i || x ||
    => || x || >= i
    for all i
    which is impossible.
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  3. #3
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    Thank you very much for your help. This solved my problem.
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