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    Showing that convergence is not uniform

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    Let fk (x) = kxe - kx, k = 1, 2, 3, ...

    It can be shown that the sequence {fk} infinity, k = 1, converges to 0

    Pointwise on [0, +oo) but convergence is not uniform on [0, +oo).

    a. Show that convergence is not uniform on [0, 1] either. Thus the sequence does not converge in the normed vector space (C ([0, 1]) , II·II infinity).

    b. Does the sequence converge to 0 in the normed vector space (C ([0, 1]) , II·II 2 (subscript 2) )?. Even though you can compute the relevant integrals manually, it may be easier to make use of a computer algebra system.

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    Solution Preview

    Problem:

    Let , k = 1, 2, 3,........
    It can be shown that the sequence {fk } infinity , k=1, converges to 0
    pointwise on [0, +oo) but convergence is not uniform on [0, +oo).
    a) Show that convergence is not uniform on [0, 1] either. Thus the sequence does not converge in the normed vector space (C ([0, 1]) , II·II infinity).
    b) Does the sequence converge to 0 in the normed vector space (C ([0, 1]) , II·II 2 (subscript 2) )?

    Solution:

    a) In order for the convergence to be uniform on [0, 1], one needs to prove that
    ...

    Solution Summary

    A pointwise convergences of uniforms are examined. The sequence converge to 0 in the normed vector spaces are determined.

    $2.49

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