Definition:Inner Product
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Definition
Let $\C$ be the field of complex numbers.
Let $\F$ be a subfield of $\C$.
Let $V$ be a vector space over $\F$.
An inner product is a mapping $\innerprod \cdot \cdot: V \times V \to \mathbb F$ that satisfies the following properties:
\((1)\) | $:$ | Conjugate Symmetry | \(\displaystyle \forall x, y \in V:\) | \(\displaystyle \quad \innerprod x y = \overline {\innerprod y x} \) | ||||
\((2)\) | $:$ | Bilinearity | \(\displaystyle \forall x, y \in V, \forall a \in \mathbb F:\) | \(\displaystyle \quad \innerprod {a x + y} z = a \innerprod x z + \innerprod y z \) | ||||
\((3)\) | $:$ | Non-Negative Definiteness | \(\displaystyle \forall x \in V:\) | \(\displaystyle \quad \innerprod x x \in \R_{\ge 0} \) | ||||
\((4)\) | $:$ | Positiveness | \(\displaystyle \forall x \in V:\) | \(\displaystyle \quad \innerprod x x = 0 \implies x = \mathbf 0_V \) |
That is, an inner product is a semi-inner product with the additional condition $(4)$.
If $\mathbb F$ is a subfield of the field of real numbers $\R$, it follows from Complex Number equals Conjugate iff Wholly Real that $\overline {\innerprod y x} = \innerprod y x$ for all $x, y \in V$.
Then $(1)$ above may be replaced by:
\((1')\) | $:$ | Symmetry | \(\displaystyle \forall x, y \in V:\) | \(\displaystyle \innerprod x y = \innerprod y x \) |
Inner Product Space
An inner product space is a vector space together with an associated inner product.
Also known as
- Innerproduct
Notation
$\innerprod x y$ is also denoted as $\left \langle {x; y} \right \rangle$.
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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: inner product