# The L^{p} Spaces

Let $(X,A,μ)$ be a measure space and let $p∈[1,∞)$. We write $L_{p}(X,A,μ)$ for the set of all measurable functions $f:X→K$ such that $∣f∣_{p}$ is integrable. For such $f$ we write

$∥f∥_{p}=(∫_{X}∣f∣_{p}dμ)_{1/p}.$

Endowed with pointwise addition and scalar multiplication $L_{p}(X,A,μ)$ becomes a vector space.

Proof We show that $L_{p}:=L_{p}(X,A,μ)$ is a linear subspace of the vector space of all $K$-valued functions on $X$. The set $L_{p}$ is nonempty since it contains the zero function. Now, suppose $f$ and $g$ are in $L_{p}$. 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

$∣f+g∣_{p}≤(∣f∣+∣g∣)_{p}≤(2max(∣f∣,∣g∣))_{p}≤2_{p}(∣f∣_{p}+∣g∣_{p}),$where $∣f∣_{p}$ and $∣g∣_{p}$ are integrable. This proves that $f+g$ lies in $L_{p}$. Finally, it is easy to see that $αf$ lies in $L_{p}$ for any scalar $α∈K$. $□$

$∥⋅∥_{p}$ is a seminorm on $L_{p}(X,A,μ)$.

Young Inequality

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

$a⋅b≤pa_{p} +qb_{q} ∀a,b≥0.$