Triangle inequality in $L^p$

Question

Consider $f_n$ as sequence of non negative functions and $f_n \in L^p(\mu),\: 1 \leq p < \infty$. I want to prove that the following holds: $$\| \sum_{k=1}^{\infty} f_k\|_p \leq \sum_{k=1}^{\infty} \|f_k\|_p $$ How can I do that?

Answer 1

Hint

$\|\cdot \|_p$ is a norm. Therefore, for all $n$, $$\left\|\sum_{k=0}^n f_k\right\|_p\leq \sum_{k=0}^n\|f_k\|_p.\tag{*}$$ Fatou's lemma allow you to conclude.

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Edit

Set $S_n=\sum_{k=0}^nf_k$ and $S_\infty =\lim_{n\to \infty }S_n$.

$$\left\|S_\infty \right\|_p=\left\|\liminf_{n\to \infty }S_n\right\|_p\underset{Fatou}{\leq} \liminf_{n\to \infty }\|S_n\|_p\underset{(*)}{\leq}\liminf_{n\to \infty }\sum_{k=0}^n\|f_k\|_p\leq \sum_{k=0}^\infty \|f_n\|_p.$$

Answer 2

The triangle inequality in $\ L^p(\mu)\ $ is known as Minkowski's inequality. You can find a proof of it in the above-linked Wikipedia article (which I have not checked for correctness), or just about any decent book on functional analysis. Once you have the simple triangle inequality, you can establish your inequality by the methods explained in the other answers.

Answer 3

The fact that you are working with the $L^p(\mu)$ norm is not important. By definition of a norm, the triangle inequality holds and specifically here it holds for nonnegative functions.

Take the definition $\lVert f_1 + f_2 \rVert_p \le \lVert f_1 \rVert_p + \lVert f_2 \rVert_p$, then take $g_2 = f_2 + f_3$ we can then use the definition again (since $f_i$ are nonnegative) to yield,

$$\lVert f_1 + g_2 \rVert_p \le \lVert f_1 \rVert_p + \lVert g_2 \rVert_p = \lVert f_1 \rVert_p + \lVert f_2 + f_3 \rVert_p \le \lVert f_1\rVert_p + \lVert f_2\rVert_p + \lVert f_3\rVert_p,$$

$$\text{i.e.} \quad \lVert f_1 + f_2 + f_3 \rVert_p \le \lVert f_1\rVert_p + \lVert f_2\rVert_p + \lVert f_3\rVert_p.$$

Then we can repeat for all $n$ by taking $g_n = f_n + f_{n+1}$ and hence we have proved the theorem.