TITLE: Why do we consider Lebesgue spaces for $p$ greater than and equal to $1$ only?
QUESTION [13 upvotes]: Why do we consider Lebesgue spaces for $p$ greater than and equal to $1$ only and not for $p$ any real number?

REPLY [4 votes]: As it is already mentioned, 
$$
\|f\|_p=\left(\int_X |f|^p\,dx\right)^{1/p},
$$
is a norm iff $p\ge 1$, since only for $p\ge 1$ satisfies the triangle inequality.
Nevertheless, the spaces $L^p(X)$, for $p\in (0,1)$ are quite interesting since they serve as the most typical example for non locally convex topological vector spaces.