Showing the triangle inequality for $||f||_p=\left(\int_{a}^{b}|f(x)|^p \, dx\right)^{1/p}$

Question

Let $a,b\in \mathbb{R}$ and $a<b$ and for $f\in C([a,b])$, $1\leq p <\infty$ let:$$||f||_p=\left(\int_{a}^{b}|f(x)|^p \, dx\right)^{1/p}$$

I want to show that the triangle inequality holds. So let $g\in C[a,b]$, then:$$||f+g||^p_p=\int_{a}^{b}|f(x)+g(x)|^p \, dx$$ I don't really know how to find a lower estimate. Do I have to use Hölder's Inequality?

Answer 1

As mentioned in the comments, what you want to prove is Minkowski's inequality and the proof that I know uses Hölder's inequality indeed. The book Functions, Spaces and Expansions does a really nice walk-through of the proof in an exercise.

1. Show that the inequality holds for $p=1$.
2. Now proceed to prove the case for $p>1$. To this end, choose $q$ such that $p^{-1}+q^{-1}=1$ and show that $q(p-1)=p$
3. Show that $|f(x)+g(x)|^p \le |f(x)| |f(x)+g(x)|^{p-1}+|g(x)||f(x)+g(x)|^{p-1}$
4. Show that $$\int_a^b |f(x)+g(x)|^p dx \le \left(\int_a^b|f(x)+g(x)|^pdx\right)^{1-1/p} \times \left[\left(\int_a^b |f(x)|^p dx\right)^{1/p} + \left(\int_a^b|g(x)|^p dx\right)^{1/p}\right]$$ using step 2 above and Hölder's inequality.

The proof should be relatively simple after step 4.