The Lebesgue Space L²(μ;ℝⁿ)

In the setting of Measurable Functions and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $n\in\mathbb{N}$. $\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$ is the set of square-integrable functions from $X$ to $\mathbb{R}^{n}$, and $f+g$, $cf$, $f\cdot g$ and the relation $\sim$ of almost-everywhere equality on $\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$ are as in Square-Integrable Vector-Valued Functions: Components, Dot Products and Almost-Everywhere Equality ($\sim$ is an equivalence relation by Square-Integrable Vector-Valued Functions: Components, Dot Products and Almost-Everywhere Equality (iv)).

(i) The space. $L^{2}(\mu;\mathbb{R}^{n})$ is the quotient $\mathcal{L}^{2}(\mu;\mathbb{R}^{n})/{\sim}$. For $f\in\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$, $[f]$ is the equivalence class of $f$ under $\sim$.

(ii) Addition and scalar multiplication. For $f,g\in\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$ and $c\in\mathbb{R}$,

$$[f]+[g]=[f+g],\qquad c[f]=[cf]$$

(well defined by Square-Integrable Vector-Valued Functions: Components, Dot Products and Almost-Everywhere Equality (iii), Square-Integrable Vector-Valued Functions: Components, Dot Products and Almost-Everywhere Equality (v), 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) Inner product. For $f,g\in\mathcal{L}^{2}(\mu;\mathbb{R}^{n})$,

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

the integral of $f\cdot g$ as an element of $\mathcal{L}^{1}(\mu)$ (well defined by Square-Integrable Vector-Valued Functions: Components, Dot Products and Almost-Everywhere Equality (ii), Square-Integrable Vector-Valued Functions: Components, Dot Products and Almost-Everywhere Equality (v) and 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), Equivalence Classes Partition the Set: Cover, Disjointness and Representatives; the Quotient Is a Set and the Canonical Projection Is a Surjection (i)).