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Isometric isomorphism

  1. Apr 23, 2014 #1
    1. The problem statement, all variables and given/known data

    Let L(l^2,l^2) be the space of bounded linear operators K:l^2->l^2.

    Now I define a map from l^infinite to L(l^2,l^2) as a->Ta(ei) to be Ta(ei)=aiei where ei is the orthonormal basic of l^2 and a=(a1,a2,...) is in l^infinte

    I want to prove this map is bijection
    can anyone give me some helps??

    2. The attempt at a solution
    I finished the part of injective but I don.t know how to show it is surjective. I tried to construct a sequence fn s.t. T(fn)=f where f(en)=fn then to show fn is indeed in l^infinite. But I failed to find such sequence
     
  2. jcsd
  3. Apr 23, 2014 #2

    micromass

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    I would try to find a counterexample.
     
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