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Extend the functional by continuity (Functional analysis)

  1. Mar 14, 2012 #1
    1. The problem statement, all variables and given/known data

    Let E be a dense linear subspace of a normed vector space X, and let Y be a
    Banach space. Suppose T0 [itex]\in[/itex] £(E, Y) is a bounded linear operator from E to Y.
    Show that T0 can be extended to T[itex]\in[/itex] £(E, Y) (by continuity) without increasing its norm.

    3. The attempt at a solution
    Someone kindly gave me a hint. I am trying to work out the details. The deadline is approaching. So I put the question here just in case. Thanks.
    For this particular problem you want to show that if (xn) converges to x then T0(xn) is a Cauchy sequence and then define f(x) as the limit of the sequence. Finally you need to show that the map is a well-defined bounded linear function.
     
  2. jcsd
  3. Mar 14, 2012 #2

    micromass

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    Use that T0 is uniform continuous.
     
  4. Mar 14, 2012 #3
    Thank you!
     
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