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Math Help - Uniform boundedness Theorem

  1. #1
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    Uniform boundedness Theorem

    Space c_0

    let y= (eta_j), eta_j element of C (complex numbers) be such that

    sum psi_i eta_j converges for every x=(psi_j0 element of c_0, where

    c_0 contained in l^{\infty} is the subspace of all complex sequences converging to zero.

    Show that sum |eta_j| < \infty. Use the Uniform Boundedness theorem.
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  2. #2
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    Let a=(a_i) be the sequence with the given properties, and let x=(x_i) be a general member of c_0.

    Consider the sequence of linear functionals T_n on c_0 given by T_nx=\sum_{i=1}^n a_ix_i.

    Using the fact that ||T_n||=\sup_{||x||=1}|T_nx|, the norm of T_n is easily shown to be ||T_n||=\sum_{i=1}^n|a_i|.

    The supremum is attained at x\in c_0 where |x_i|=1 with a_ix_i=|a_i| for 1\leq i\leq n, and where x_i=0 for i>n. Note that x\in c_0 and ||x||=\sup |x_i|=1.

    Now \sup_n |T_nx|<\infty for all x\in c_0, since by hypothesis the series \sum_{i=1}^\infty a_ix_i converges for all such x.

    By the uniform boundedness principle, \sup_n||T_n||<\infty. Therefore \sum_{i=1}^\infty|a_i|<\infty, which means that a\in l^1.
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