The L^p Spaces

Let $(X,\mathcal{A},\mu )$ be a measure space and let $p\in [1,\infty )$. We write ${\mathcal{L}}^p(X,\mathcal{A},\mu )$ for the set of all measurable functions $f:X\to \mathbb{K}$ such that $|f{|}^p$ is integrable. For such $f$ we write

$$\Vert f{\Vert }_p={({\int }_X|f{|}^{p}d\mu )}^{1/p}.$$

Endowed with pointwise addition and scalar multiplication ${\mathcal{L}}^p(X,\mathcal{A},\mu )$ becomes a vector space.

Proof   We show that ${\mathcal{L}}^p:={\mathcal{L}}^p(X,\mathcal{A},\mu )$ is a linear subspace of the vector space of all $\mathbb{K}$-valued functions on $X$. The set ${\mathcal{L}}^p$ is nonempty since it contains the zero function. Now, suppose $f$ and $g$ are in ${\mathcal{L}}^p$. Then the sum $f+g$ is measurable, because $f$ and $g$ are measurable. Moreover, the function $|f+g{|}^p$ is integrable, because we have the estimate

$$|f+g{|}^p\le (|f|+|g|{)}^p\le (2\max (|f|,|g|){)}^p\le 2^p(|f{|}^p+|g{|}^p),$$

where $|f{|}^p$ and $|g{|}^p$ are integrable. This proves that $f+g$ lies in ${\mathcal{L}}^p$. Finally, it is easy to see that $\alpha f$ lies in ${\mathcal{L}}^p$ for any scalar $\alpha \in \mathbb{K}$. $\square$

$\Vert \cdot {\Vert }_p$ is a seminorm on ${\mathcal{L}}^p(X,\mathcal{A},\mu )$.

Young Inequality

Consider $p,q>1$ such that $1/p+1/q=1$. Then

$$a\cdot b\le \frac{a^p}{p}+\frac{b^q}{q}\forall a,b\ge 0.$$