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Math Help - How to show that something is a subspace of something else?

  1. #1
    Ase
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    How to show that something is a subspace of something else?

    Hi

    I know that this is a basic question but it is one that I cannot seem to be able to get my head around. If I have a set



    and another set where the thing condition that there can only be finitely many x_k different from zero is replaced by the condition that the supremum of x_k must be finite.

    How can I show that the second set is a subspace of the first set?

    I know about the addition and multiplication with scalar, but I just cannot seem to be able to understand it so I was hoping for a thorough example.

    Thanks a lot
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Ase View Post
    Hi

    I know that this is a basic question but it is one that I cannot seem to be able to get my head around. If I have a set



    and another set where the thing condition that there can only be finitely many x_k different from zero is replaced by the condition that the supremum of x_k must be finite.

    How can I show that the second set is a subspace of the first set?

    I know about the addition and multiplication with scalar, but I just cannot seem to be able to understand it so I was hoping for a thorough example.

    Thanks a lot
    The first thing you need to show is that the first set is a subset of the second. That is true because, if there are only a finite number of non-zero x_ks, there is a largest one which is the finite supremum.

    The second thing you must show is that the set is closed under addition and scalar multiplication. I am assuming "coordinate-wise" addition and scalar multiplication here: \{x_k\}+ \{y_k\}= \{x_k+ y_k\} and  r\{x_k\}= \{rx_k\}. Scalar multiplication is easy: all but a finite number of the x_k are 0 and any scalar times 0 is 0. So all but a finite number of rx_k are 0.

    For addition, suppose \{x_k\} has n non-zero entries and \{y_k\} has m non-zero entries. Can you see that \{x_k+ y_k\} has, at most, m+n non-zero entries?
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  3. #3
    Ase
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    Again you are a lifesaver HallsofIvy. I have been trying to get people to explain it to me in ways in understand so many times. The only difference here is that you have made it understandable.

    Thanks a million
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