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...cal objects belong to—or are associated with—some [[linear space]] with [[inner product]]. The algebraic form of a physical quantity is—explicitly or implicitly� ...ap of the Cartesian product ℝ³×ℝ³ into ℝ). The inner product satisfies

{{r|Inner product space}}

{{r|Inner product}}

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{{r|Inner product space}}

...space]] over the sub-field <math>F</math> of real or complex numbers with inner product <math>\langle \cdot,\cdot \rangle</math>, and let <math>x_1,x_2,\ldots,x_n<

This relation can be immediately generalized to ''m''-dimensional inner product spaces. Let the space ''V''_''m'' allow an orthogonal direct sum ...'-dimensional linear subspace of a linear space of dimension ''m''.) The inner product of two vectors ''v'' and ''w'' is notated as (''v'', ''w''), which is comm

{{r|Inner product}}

...ite inner product between the elements &psi; and &phi; of the same complex inner product space. We follow the physical convention

...[[eigenvector]]s, [[linear operators]], [[matrix representation]]s, and [[inner product]]s find broad application in [[science]].

{{r|Inner product}}

The distance is defined by means of the following positive definite [[inner product]] on ℝ^''n'', ...b>i </sub> of '''y'''. Further, &lang;'''a''', '''b'''&rang; stands for an inner product between '''a''' and '''b'''.

{{r|Inner product space}}

{{r|Inner product}}

{{r|Inner product space}}

Formally, the Euclidean plane is a 2-dimensional [[affine space]] with [[inner product]].

...que complements each of the other). However, if ''V'' is in addition an [[inner product]] space, then there is a unique ''orthogonal complement''

* [[Inner product space]]

...c bracket, <math>\langle\cdot|\cdot\rangle</math>, which is equal to the [[inner product]] between them.

...oduct is usually notated as a bra and ket, following [[Dirac]]. Thus, the inner product of &Phi; and &Psi; is written as, ...expressed by stating that &Phi; and &Phi;&prime; are orthogonal (have zero inner product).