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Math Help - Infinite Dimensional Vector Spaces

  1. #1
    Senior Member Sampras's Avatar
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    Infinite Dimensional Vector Spaces

    Suppose you have a infinite dimensional vector space  V . Suppose  T \in \mathcal{L}(V,W) . Can we have  \dim V = \dim \text{null} \ T + \dim \text{range} \ T?

    E.g. can we partition an infinite dimensional vector space into finite dimensional ones?
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    I'm not sure how your two questions are related. The dimension theorem holds for infinite-dimensional vector spaces; but all that this tells us is that if \dim V = \infty then either the range or the kernel of T must be infinite-dimensional.

    It is impossible to partition a vector space into subspaces. If W_1, W_2 are subspaces and W_1 \cup W_2 = V then we must have W_1 \subset V or W_2 \subset V. On top of that, the zero vector would have to be in every part.
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