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Math Help - Banach Space problem

  1. #1
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    Banach Space problem

    Let X be the space of complex polynomials on the unit interval [0,1] regarded as a subspace of the complex Banach space C([0,1]) of complex valued functions on [0,1] equipped with the supremum norm. For j=0,1,2,... let f^j be the element of X defined, for t in [0,1] by

    (f^j)(t)= t^j.

    Define the mapping T from X to itself, for j= 0,1,2 by Tf^j= (1/j+1)f^j, and elsewhere by linearity. By expressing T as an integral, or otherwise, prove that T is in B(X), T is of norm 1, T maps X one to one onto X, but that the algebraic inverse of T in not in B(X) (where B(X) is the set of bounded linear operators from X into itself).

    I can't get started with this question, because I don't know what integral they're looking for. How should T be expressed?

    Many thanks.
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  2. #2
    MHF Contributor

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    How can you express x\mapsto \frac{1}{n+1}x^n in terms of x\mapsto x^n, linearly, using an integral? Simply by \frac{1}{n+1}x^n = \frac{1}{x}\int_0^x t^n dt. Hence in general TP is the polynomial given by (TP)(x)=\frac{1}{x}\int_0^x P(t) dt. (Linear, and coincides with the values chosen for P(x)=x^n).

    Now that you should be started, I let you try the next questions by yourself. Ask for additional help if you need (telling us what you tried, what failed).
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  3. #3
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    Thanks very much. I can now everything except that T is onto, and of norm one. For T onto, is it best to use the original definition of (T(f^j))(t)= (1/j+1)f^j, or the integral definition.

    Also, for the norm bit, I can show that the norm of T is less than or equal to one, but not greater than or equal to.
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