Unfortunately I never had an opportunity to take a functional analysis/PDE module so I keep running into steps I'm unsure of when I'm looking at proofs. I'm trying to fix that now with some books so apologies in advance if these questions seem trivial but it would be great to clear up this stuff once and for all.
In Brezis book on PDEs in theorem 4.7 he shows that $\Vert \cdot \Vert_p$ is a norm for $1 \le p \le \infty$. I'm not sure of some of the inequalities and statements he uses. Let $f, g \in L^p$. We have $$|f + g|^p \le (|f| + |g|)^p \le 2^p(|f|^p + |g|^p).$$
Therefore $f + g \in L^p$.
1. What inequality is being used in the last step here?
2. why does this statement imply $f + g \in L^p$?
$$\Vert f + g \Vert_p^p = \int |f + g|^{p-1}|f + g| \le \int |f + g|^{p-1}|f| + \int |f + g|^{p - 1}|g|.$$ But $|f + g|^{p-1} \in L^{p\prime}$ and by Holder's inequality we obtain
$$\Vert f + g \Vert_p^p \le \Vert f + g \Vert_p^{p-1}(\Vert f \Vert_p + \Vert g \Vert_p).$$
That is, $\Vert f + g \Vert_p \le \Vert f \Vert_p + \Vert g \Vert_p.$
3. How do we know $|f + g|^{p-1} \in L^{p\prime}$? If some function $a \in L^p$ then $|a|^{p-1}$ is in $L^{p\prime}$...is that it?
4. Why does this final statement imply $\Vert f + g \Vert_p \le \Vert f \Vert_p + \Vert g \Vert_p$
1) If $|f| \ge |g|$ then $$(|f| + |g|)^p \le (2|f|)^p = 2^p |f|^p \le 2^p (|f|^p + |g|^p).$$ You can derive the same inequality if $|g| \le |f|$.
2) Once the inequality in 1) is established you have $$\int |f+g|^p \le 2^p \int |f|^p + 2^p \int |g|^p < \infty.$$ Thus by definition $f+g \in L^p$.
3) Since $p' = \frac{p}{p-1}$ it follows that $$ (|f+g|^{p-1})^{p'}= |f+g|^p.$$ If one side is integrable so is the other, so that $|f+g|^{p-1} \in L^{p'}$ if and only if $|f+g| \in L^p$.
4) If $\|f+g\|_p = 0$ the inequality is trivial; otherwise divide both sides by $\|f+g\|_p^{p-1}$.