Definition:Inner Product

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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 $\map {\LL^2} {X, \Sigma, \mu}$ 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 defined as

Some texts define an inner product only for vector spaces over $\R$ or $\C$.

This ensures that for all $v \in V$, the inner product norm:

$\norm v = \sqrt {\innerprod v v}$

is a scalar.

$\mathsf{Pr} \infty \mathsf{fWiki}$ prefers the more general definition, and lists additional requirement on $\GF$ in theorems where it is needed, such as the Gram-Schmidt Orthogonalization theorem.


Some sources use the term inner product to refer merely to the dot product, which is in fact a specific example of an inner product.


Also denoted as

The inner product $\innerprod x y$ can also be 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.


Also known as

Some texts refer to an inner product $\innerprod \cdot \cdot$ as a scalar product

As this term is ambiguous, it is not used in this context by $\mathsf{Pr} \infty \mathsf{fWiki}$.

Some texts refer to $\innerprod \cdot \cdot$ as an innerproduct.


Also see

  • Results about inner products can be found here.


Sources

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