# Definition:Inner Product

## Definition

### Complex Inner Product

Let $V$ be a vector space over a complex subfield $\GF$.

A (complex) inner product is a mapping $\innerprod \cdot \cdot: V \times V \to \GF$ that satisfies the complex inner product axioms:

 $(1)$ $:$ Conjugate Symmetry $\ds \forall x, y \in V:$ $\ds \quad \innerprod x y = \overline {\innerprod y x}$ $(2)$ $:$ Linearity in first argument $\ds \forall x, y \in V, \forall a \in \GF:$ $\ds \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z$ $(3)$ $:$ Non-Negative Definiteness $\ds \forall x \in V:$ $\ds \quad \innerprod x x \in \R_{\ge 0}$ $(4)$ $:$ Positiveness $\ds \forall x \in V:$ $\ds \quad \innerprod x x = 0 \implies x = \mathbf 0_V$

That is, a (complex) inner product is a complex semi-inner product with the additional condition $(4)$.

### Real Inner Product

Let $V$ be a vector space over a real subfield $\GF$.

A (real) inner product is a mapping $\innerprod \cdot \cdot: V \times V \to \GF$ that satisfies the real inner product axioms:

 $(1')$ $:$ Symmetry $\ds \forall x, y \in V:$ $\ds \innerprod x y = \innerprod y x$ $(2)$ $:$ Linearity in first argument $\ds \forall x, y \in V, \forall a \in \GF:$ $\ds \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z$ $(3)$ $:$ Non-Negative Definiteness $\ds \forall x \in V:$ $\ds \quad \innerprod x x \in \R_{\ge 0}$ $(4)$ $:$ Positiveness $\ds \forall x \in V:$ $\ds \quad \innerprod x x = 0 \implies x = \mathbf 0_V$

That is, a (real) inner product is a real semi-inner product with the additional condition $(4)$.

### Inner Product Space

An inner product space is a vector space together with an associated inner product.

## Examples

### 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 $\map {\mathcal L^2} {X, \Sigma, \mu}$ 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 $g$.

• Innerproduct

## Also denoted as

$\innerprod x y$ is also denoted as $\left \langle {x; y} \right \rangle$.

If there is more than one vector space under consideration, then the notation $\innerprod x y_V$ for a vector space $V$ is commonplace.