# Minkowski inequality of infinite sum

For $1\leq p <\infty,$
Given $\{f_n\}^{\infty}_{n=1}$ be a sequence of function in $L^{p}(\mathbb{R}).$
Show that $\left\Vert \sum\limits_{n=1}^\infty f_n\right\Vert_p \leq \sum\limits_{n=1}^\infty \left\Vert f_n\right\Vert_p$.
(Minkowski inequality can be used.)

The hint is too use monotone and dominated convergence theorem

Clearly, $\left\Vert \sum\limits_{n=1}^m f_n\right\Vert_p \leq \sum\limits_{n=1}^m \left\Vert f_n\right\Vert_p$ $\forall m \in\mathbb{N}$.
The question is how can I pass the limit into the norm on the left.

More precisely, how to use the monotone and dominated convergence theorem? What is the dominate function?

Consider using Fatou's lemma instead. Let $s_n := \left|\sum\limits_{k = 1}^n f_k\right|^p$. By Fatou's lemma, $$\left\|\sum_{n = 1}^\infty f_n\right\|_p^p = \|\lim_{n\to \infty} s_n\|_1 \le \varliminf_{n\to \infty} \|s_n\|_1 = \varliminf_{n\to \infty} \left\|\sum_{k = 1}^n f_k\right\|_p^p.$$ Therefore $$\left\|\sum_{n = 1}^\infty f_n\right\|_p \le \varliminf_{n\to \infty} \left\|\sum_{k = 1}^n f_k\right\|_p.$$ By Minkowski's inequlity,

$$\varliminf_{n\to \infty} \left\|\sum_{k = 1}^n f_k\right\|_p \le \varliminf_{n\to \infty} \sum_{k = 1}^n \|f_k\|_p = \sum_{n = 1}^\infty \|f_n\|_p.$$

Hence

$$\left\|\sum_{n = 1}^\infty f_n\right\|_p \le \sum_{n = 1}^\infty \|f_n\|_p.$$

• Probably a long shot since this was so long ago, but could you explain how you switch to the 1 norm there Jun 18, 2022 at 19:47
• Hi @TdotA, note that $\|f\|_p^p = \| |f|^p\|_1$ since $\|f\|_p^p = \int |f|^p$. This fact was used in the step you mentioned.
– kobe
Jun 18, 2022 at 20:02

Kobe's answer is much more efficient, but if you want to use MCT and DCT, I believe this works.

I assume you can show the result if $$\int \lim_{k\to\infty}\left|\sum_{n=1}^k f_n\right|^p = \lim_{k\to\infty}\int \left|\sum_{n=1}^k f_n \right|^p .$$So we need to justifty switching the order of limits, which amounts to showing that the sequence of functions $|h_k|^p = |\sum_{n=1}^kf_n|^p \leq g^p$ for some $g^p \in L^1$ and all $k \in \mathbb N$.

First, notice that if $\sum_{n=1}^\infty \Vert f_n \Vert_p = \infty$, then the result holds trivially. So, you can suppose that $\sum_{n=1}^\infty \Vert f_n \Vert_p = B < \infty$ for some real number $B \geq 0$. Define $g_k = \sum_{n=1}^k |f_n|$ and $g = \sum_{n=1}^\infty |f_n|$, and notice that $\Vert g_k \Vert_p \leq \sum_{n=1}^\infty \Vert f_n\Vert_p = B < \infty$ by Minkowski's Inequality. By the montone convergence theorem, since $g_k^p$ is an increasing sequence of non-negative functions that coverges to to $g^p$ ($g_k$ increases to $g$), we have $$\int g^p = \lim_{k\to\infty} \int g_k^p = \lim_{k\to\infty} \Vert g_k\Vert_p^p \leq B^p < \infty$$ so that $g^p \in L^1$. Also, it follows that $g(x) = \sum_{n=1}^\infty |f_n(x)| < \infty$ for almost every $x$. Thus, for almost every $x$, we know that $g(x)$ absolutely converges, and thus converges for almost every $x$ (since absolutely convergent series of real numbers converge). That is, $h(x) = \sum_{n=1}^\infty f_n(x) < \infty$ for almost every $x$. Lastly, if we let $h_k = \sum_{n=1}^k f_n$, then we have $|h_k|^p \leq g^p$ where $g^p \in L^1$.