$(\Omega, \mathcal{F}, u)$ is a measure space, and $L^p(\Omega, \mathcal{F}, u)$ is its $L^p$ space. Define $N_p(f) = \int_{\Omega} |f|^p\, d\mu$. $\forall f, g \in L^p(\Omega, \mathcal{F}, u)$,
- when $0
, $N_p(f+g) \leq N_p(f)+N_p(g) $ is true according to Wikipedia. This can help to show that $L^p(\Omega, \mathcal{F}, u)$ is a vector space. I was wondering how to prove the inequality is true?
when $p \geq 1$, is the inequality $N_p(f+g) \leq N_p(f)+N_p(g) $ still true?
If not,
(1) can the inequality be modified to be true? Note I am not asking about the triangle inequality of $L^p$ norm.
(2) how can one show that $L^p(\Omega, \mathcal{F}, u)$ is a vector space?
Thanks and regards!