# Sum of two Uniform Convergent Functional Series

• Jan 22nd 2011, 04:43 AM
garunas
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
• Jan 22nd 2011, 05:27 AM
FernandoRevilla
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

$n\geq n_o$