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Math Help - Completeness

  1. #1
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    Completeness

    I was looking at the following proof that the quotient space X/M where X is a Banach space and M is a closed subspace:

    PlanetMath: quotients of Banach spaces by closed subspaces are Banach spaces under the quotient norm

    Can someone elaborate for me on the second to last line beginning x-s_k+M? In particular I'm shaky on the last two equalities.

    Thanks.
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  2. #2
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    In a quotient space, the coset containing x+y is (by definition) the sum of the coset containing x and the coset containing y. In other words, (x+y) + M = (x+M) + (y+M). By induction, this extends to any finite sum of cosets.

    That is all that is happening in the line x-s_k+M = (x+M) - (s_k+M) = (x+M) - \sum_{n=1}^k(x_n+M) = (x+M) - \sum_{n=1}^kX_n. The first two equalities are using that fact about sums of cosets (remember that s_k = \textstyle\sum_{n=1}^k x_n), and the last equality comes from the definition of X_n = x_n+M.
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