The triangle inequality of the $L^p$-norm for a infinite sum

Question

I just wonder that $\|\sum _{i=1}^\infty f_i\|_p\leq\sum _{i=1}^\infty\|f_i\|_p$.

Where $\|.\|_p$ is the $L^p$- norm ,that is, $\|f\|_p=(\int_U|f|^pdm)^{1/p}$

Does this always work, or under what conditions?

Answer 1

This works in any normed space, but it requires the assumption that $\sum_{i=1}^\infty f_i$ converges, because otherwise the left side is not defined.

Let $(X,\|\cdot\|)$ be a normed space and let $\sum_{i=1}^\infty x_i$ be a convergent series in $X$. For each $n\in\mathbb{N}$, let $s_n=\sum_{i=1}^n x_i$. By definition, convergence means that $\lim_n s_n$ exists. Let $x=\lim_n s_n=\sum_{i=1}^\infty x_i$. Then, again by definition, $\lim_n \|x-s_n\|=0$. By the reverse triangle inequality, $|\|x\|-\|s_n\||\leqslant \|x-s_n\|$ for all $n$, so $$0\leqslant \lim_n |\|x\|-\|s_n\||\leqslant \lim_n \|x-s_n\|=0,$$ and $\lim_n |\|x\|-\|s_n\||=0$. This means that $|\sum_{i=1}^\infty x_i|=|x|=\lim_n |s_n|.$$ (We're just using the fact that the norm is continuous, which follows from the reverse triangle inequality).

By the usual triangle inequality, we know that for each $n$, $$\|s_n\|=\|\sum_{i=1}^n x_i\|\leqslant \sum_{i=1}^n\|x_i\|\leqslant \sum_{i=1}^\infty \|x_i\|.$$ Note that we did not assume that $\sum_{i=1}^\infty \|x_i\|$ is finite. The inequality is trivially true if the right hand side is $\infty$.

Therefore we have $$\|x\|=\lim_n \|s_n\|\leqslant \sum_{i=1}^\infty \|x_i\|.$$