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 $g$.
Also known as
Some texts refer to $\innerprod \cdot \cdot$ as a scalar product
As this term is disambiguous, it is not used by $\mathsf{Pr} \infty \mathsf{fWiki}$.
Some texts refer to $\innerprod \cdot \cdot$ as an innerproduct.
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 $\Bbb F$ in theorems where it is needed, such as the Gram-Schmidt Orthogonalization theorem.
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.
Also see
- Definition:Semi-Inner Product, a slightly more general concept.
- The most well-known example of an inner product is the dot product (see Dot Product is Inner Product).
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples: Definition $1.1$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): inner product
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Definitions
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $8.1$: Inner Products