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Math Help - Sum of two Uniform Convergent Functional Series

  1. #1
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    Question Sum of two Uniform Convergent Functional Series

    Let F(x):=\displatstyle\sum\limits_{j=1}^{\infty}f_{j}  (x) and
    G(x):=\displatstyle\sum\limits_{j=1}^{\infty}g_{j}  (x)

    be uniformly convergent functional series.

    Show that \displatstyle\sum\limits_{j=1}^{\infty}(f_{j} + g_{j}) is uniformly converget to F + G

    Now I suppose I need to use the Chebyshev Norm and show that \bigg\lVert\displatstyle\sum\limits_{j=1}^{\infty}  (f_{j} + g_{j}) - (F + G)\bigg\rVert_{\infty} <\epsilon

    Or any other method perhaps??
    I have no idea other than that one, any help would be much appriciated thanks
    Last edited by garunas; January 22nd 2011 at 05:44 AM. Reason: unfinised
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    You only need the standard argument:

    \left |{\displaystyle\sum_{j=1}^n{(f_j(x)+g_j(x))-(F(x)+G(x))}}\right |\leq

    \left |{\displaystyle\sum_{j=1}^n{f_j(x)-F(x))}}\right |+\left |{\displaystyle\sum_{j=1}^n{(g_j(x)-G(x))}}\right |< \dfrac{ \epsilon}{2}+\dfrac{ \epsilon}{2}= \epsilon

    for every

    x\in D(F)\cap D(G)

    and for

    n\geq n_o

    adequately chosen.


    Fernando Revilla
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