Lp Space

Definition §

Given some $p\in \mathbb{R}$ where $p\ge 1$, we define the space $L^p[a,b]=\left\{f\mid {\int }_a^b|f(x){|}^{p}dx<\infty \right\}$. This space is a linear space, i.e. its elements are closed under linear combination.

The $L^p$ space is also equipped with a canonical norm, the $L^p$ norm.

Remarks §

1. If $1\le p\le q<\infty$, then $L^q[a,b]\subset L^p[a,b]$.