Minkowski inequality of infinite sum

Question

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?

Answer 1

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

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

Answer 2

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 $g^p \in L^1$.