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Math Help - finite spanning set<->frame

  1. #1
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    finite spanning set<->frame

    Show that every finite spanning set for \mathbb{R}^d is a frame for \mathbb{R}^d
    (frame: \exists A,B >0 \in  such that A ||x||^2<=\sum|<x,f_k>|^2<=B||x||^2 )

    and that every finite frame {f_1,\ldots,f_n}\subset \mathbb{R}^d, n>d is a spanning set for \mathbb{R}^d
    Last edited by alighieri; March 4th 2010 at 05:18 AM.
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  2. #2
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    finite frame->spanning

    Assume  {f_1,\ldots,f_n} is a frame, but not a spanning set \rightarrow nontrivial null space for the matrix with rows f_k\rightarrow\exists x such that ||x||>0 but \forall k, |<x,f_k>|=0\rightarrow Contradiction of framing condition for any postive A
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  3. #3
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    Thanks. Also, half of the other direction is easy:

    \sum |<x,f_k>|^2\leq\sum ||x||^2||f_k||^2 by Cauchy-Schwartz =||x||^2\sum||f_k||^2 and spanning \rightarrow \exists ksuch that ||f_k||>0\rightarrow B=\sum||f_k||^2
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