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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 Summary

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

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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
...

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