Inner Product/Examples

Examples of Inner Products

Sequences with Finite Support

Let $\GF$ be either $\R$ or $\C$.

Let $V$ be the vector space of sequences with finite support over $\GF$.

Let $f: \N \to \R_{>0}$ be a mapping.

Let $\innerprod \cdot \cdot: V \times V \to \GF$ be the mapping defined by:

$\ds \innerprod {\sequence {a_n} } {\sequence {b_n} } = \sum_{n \mathop = 1}^\infty \map f n a_n \overline{ b_n }$

Then $\innerprod \cdot \cdot$ is an inner product on $V$.

Inner Product on $L^2$ Space

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 $g$.