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Math Help - Infinite Dimensional Vector Space.

  1. #1
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    Infinite Dimensional Vector Space.

    There is a 3 part question that I have been working on.

    The first 2 parts are show that for a finite dimensional vector space:

    S \circ T is invertible if and only if S and T are invertible.
    S \circ T = I if and only if T \circ S = I

    Theese two I have successfully proven. It is the last part I am having trouble with which is:

    Give an example for each that shows that the statements are false for infinite dimensional vector spaces.

    Any help would be appreciated, Thanks!
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Hint : consider the vector space V of sequences of real numbers with the map S: (a_1,\dots,a_n,\dots) \mapsto (0,a_1,\dots,a_n,\dots). Show that S has a left inverse which is not a right inverse.
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  3. #3
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    Thanks for the hint!

    Here is what I got

    Let
    and T: (b_1,...,b_n,...) \mapsto (b_2,...,b_{n+1},...)

    S \circ T will give us the original sequence however
    T \circ S wont. This proves the second
    " if and only if " does not hold for infinite dimensional matrices.

    However Im still not sure how to prove the first part " is invertible if and only if and are invertible." does not hold for infinite dimensional matrices.
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    Well you have to be careful; T\circ S is the identity map, not S\circ T. You are right that it proves the second part of the problem.

    For the first part, notice that T\circ S is invertible (it's the identity!) but that T is not. So the part of the statement which says "If T\circ S is invertible, then both T and S are invertible" is false. The part which says "If both T and S are invertible, then T\circ S is invertible" is always true; the inverse of T\circ S is then given by S^{-1}\circ T^{-1}.
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