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Math Help - Finite Dimensional

  1. #1
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    Finite Dimensional

    I am having a fair amount of trouble with what would appear to be a very straight forward question. The question states:

    Show that the set consists of all real sequences converging to zero is a vector space and show that it is not finite dimensional.

    If anyone could give me a rough idea how to get started on this question or how to approach it, I would be very appreciative. Thanks.

    Edit: I should also point out that the english is very bad lol, it should probably read something like: Show that the set "which" consists ....
    Last edited by GreenDay14; September 23rd 2009 at 08:13 AM.
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  2. #2
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    pointers on how I tried

    Part 1: If X,Y are real sequences converging to 0, it is easy to show aX + bY, where a,b are real will also converge to 0. So set of S of all such seqences is a vector space.

    Part 2: Attempt by contradiction. Assume the space is finite dimensional, get a basis and then construct a sequence which though converges to 0, cannot be a expressed as a linear combination of the basis we selected earlier. Hence proving the space is infinite dimensional.

    As a further hint any n-dimensions vector space is isomorphic to \Re^n

    Hope it helps
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  3. #3
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    I am trying to figure it out and I understand how it is not finite dimensional but how can i prove it is a vector space when one of the properties of a vector space is that it must contain the zero vector. However, the question is stating that it is converging to zero, so wouldn't that mean that there couldn't be a zero vector since it is only converging?
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  4. #4
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    No - {0,0,0,0,0,0,0,0,0,0,0,0.......................} by all means converges to 0. It perfectly satisfies the definition of a seq converging to 0
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  5. #5
    MHF Contributor Bruno J.'s Avatar
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    The sequence (0,0,0,...) converges to 0 trivially.

    To solve the other part of the problem, note that for any n, any sequence of the form (x_1,...,x_n,0,0,0,..) is admissible, where the x_i are any numbers you wish. In other words, for any n your vector space contains a subspace isomorphic to \mathbb{R}^n, so it can't be finite-dimensional.
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  6. #6
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    Thank you both. I greatly appreciate the help.
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