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Thread: Sum of two Uniform Convergent Functional Series

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

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

    be uniformly convergent functional series.

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

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

    Or any other method perhaps??
    I have no idea other than that one, any help would be much appriciated thanks
    Last edited by garunas; Jan 22nd 2011 at 04:44 AM. Reason: unfinised
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    You only need the standard argument:

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

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

    for every

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

    and for

    $\displaystyle n\geq n_o$

    adequately chosen.


    Fernando Revilla
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