Definition:L-2 Inner Product

Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\map {\LL^2} {X, \Sigma, \mu}$ be the Lebesgue $2$-space of $\struct {X, \Sigma, \mu}$.

Let $\map {L^2} {X, \Sigma, \mu}$ be the $L^2$ space of $\struct {X, \Sigma, \mu}$.

We define the $L^2$ inner product $\innerprod \cdot \cdot : \map {L^2} {X, \Sigma, \mu} \times \map {L^2} {X, \Sigma, \mu} \to \R$ by:


 * $\ds \innerprod {\eqclass f \sim} {\eqclass g \sim} = \int \paren {f \cdot g} \rd \mu$

where:


 * $\eqclass f \sim, \eqclass g \sim \in \map {L^2} {X, \Sigma, \mu}$ where $\eqclass f \sim$ and $\eqclass g \sim$ are the equivalence class of $f, g \in \map {\LL^2} {X, \Sigma, \mu}$ under the $\mu$-almost everywhere equality relation.
 * $\ds \int \cdot \rd \mu$ denotes the usual $\mu$-integral of $\mu$-integrable function
 * $f \cdot g$ denotes the pointwise product of $f$ and $g$.

Also see

 * $L^2$ Inner Product is Well-Defined
 * $L^2$ Inner Product is Inner Product
 * $L^2$ Space forms Hilbert Space