Definition:Inner Product/Complex Field
Definition
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 $(4)$.
Complex Inner Product Space
Let $V$ be a vector space over a complex subfield $\GF$.
Let $\innerprod \cdot \cdot : V \times V \to \GF$ be an complex inner product on $V$.
We say that $\struct {V, \innerprod \cdot \cdot}$ is a (complex) inner product space.
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:Complex 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).
- Inner Product/Examples for examples of both complex inner products and real inner products
Sources
This page may be the result of a refactoring operation. As such, the following source works, along with any process flow, will need to be reviewed. When this has been completed, the citation of that source work (if it is appropriate that it stay on this page) is to be placed above this message, into the usual chronological ordering. If you have access to any of these works, then you are invited to review this list, and make any necessary corrections. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{SourceReview}} from the code. |
- 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