The Lebesgue Space Lᵖ(μ)

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $p\ge1$ be a real number. $\mathcal{L}^{p}(\mu)$ is the set of $p$-integrable functions, $\|f\|_{p}$ is the $p$-seminorm of $f\in\mathcal{L}^{p}(\mu)$, and $\sim$ is the relation of almost-everywhere equality on $\mathcal{L}^{p}(\mu)$ of Almost-Everywhere Equality on the p-Integrable Functions, an equivalence relation by Almost-Everywhere Equality on the p-Integrable Functions (i).

(i) The Lebesgue space. The Lebesgue space $L^{p}(\mu)=L^{p}(X,\mathcal{F},\mu)$ is the quotient $\mathcal{L}^{p}(\mu)/{\sim}$. For $f\in\mathcal{L}^{p}(\mu)$, $[f]$ is the equivalence class of $f$ under $\sim$.

(ii) Addition and scalar multiplication. The sum of two elements of $L^{p}(\mu)$ and the multiple of an element of $L^{p}(\mu)$ by a real number are given by

$$[f]+[g]=[f+g],\qquad c[f]=[cf]\qquad(f,g\in\mathcal{L}^{p}(\mu),\ c\in\mathbb{R})$$

(a binary operation on $L^{p}(\mu)$ and a function $\mathbb{R}\times L^{p}(\mu)\to L^{p}(\mu)$, well defined by Minkowski's Inequality (i), Almost-Everywhere Equality on the p-Integrable Functions (ii), Compatible Operations, Maps and Relations Pass to the Quotient (i), and Compatible Operations, Maps and Relations Pass to the Quotient (ii) applied to $f\mapsto cf$ for each $c$).

(iii) The norm. The norm of $[f]$, for $f\in\mathcal{L}^{p}(\mu)$, is

$$\|[f]\|_{p}=\|f\|_{p}$$

(a function $L^{p}(\mu)\to\mathbb{R}$, well defined by Almost-Everywhere Equality on the p-Integrable Functions (iii) and A Map Constant on Equivalence Classes Factors Uniquely through the Quotient (i)).

(iv) The inner product. If $p=2$, the inner product of $[f]$ and $[g]$, for $f,g\in\mathcal{L}^{2}(\mu)$, is

$$\langle[f],[g]\rangle=\int fg\,d\mu,$$

the integral of $fg\in\mathcal{L}^{1}(\mu)$ (a function $L^{2}(\mu)\times L^{2}(\mu)\to\mathbb{R}$, well defined by Almost-Everywhere Equality on the p-Integrable Functions (iv), Equivalence Classes Partition the Set: Cover, Disjointness and Representatives; the Quotient Is a Set and the Canonical Projection Is a Surjection (iv), Equivalence Classes Partition the Set: Cover, Disjointness and Representatives; the Quotient Is a Set and the Canonical Projection Is a Surjection (v) and Equivalence Classes Partition the Set: Cover, Disjointness and Representatives; the Quotient Is a Set and the Canonical Projection Is a Surjection (i)).