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- ...tors. The projection operation is a powerful geometric tool that makes the inner product a desirable convenience, especially for the purposes of [[optimization (mat ==Formal definition of inner product==3 KB (511 words) - 00:25, 20 February 2010
- 135 bytes (19 words) - 15:24, 28 November 2008
- 170 bytes (21 words) - 15:29, 28 November 2008
- ...rm on the vector space on which it is defined. A [[completeness|complete]] inner product space is called a [[Hilbert space]]. ==Examples of inner product spaces==1 KB (204 words) - 14:38, 4 January 2009
- 12 bytes (1 word) - 21:15, 6 October 2007
- 12 bytes (1 word) - 21:23, 6 October 2007
- {{r|Inner product space}}905 bytes (145 words) - 15:27, 28 November 2008
- A vector space that is endowed with an inner product and the corresponding norm.116 bytes (17 words) - 13:40, 4 January 2009
- {{r|Inner product}}940 bytes (149 words) - 15:13, 28 July 2009
Page text matches
- * [[inner product]], a generalisation of the above in an abstract vector space.242 bytes (36 words) - 12:38, 31 May 2009
- {{r|Inner product}}798 bytes (103 words) - 16:04, 11 January 2010
- ...rm on the vector space on which it is defined. A [[completeness|complete]] inner product space is called a [[Hilbert space]]. ==Examples of inner product spaces==1 KB (204 words) - 14:38, 4 January 2009
- {{r|Inner product space}}338 bytes (42 words) - 19:08, 5 October 2009
- ...n two vectors (resulting into a vector) and the symbol "•" stands for an [[inner product]] of two vectors (resulting into a scalar).519 bytes (83 words) - 03:13, 15 January 2009
- {{r|Inner product space}} {{r|Inner product}}366 bytes (48 words) - 11:31, 27 July 2008
- A complete inner product space.67 bytes (8 words) - 12:32, 4 January 2009
- ...tors. The projection operation is a powerful geometric tool that makes the inner product a desirable convenience, especially for the purposes of [[optimization (mat ==Formal definition of inner product==3 KB (511 words) - 00:25, 20 February 2010
- ..." definition for it. In fact, in this case the contrary is true: it is the inner product which is defined directly while the notion of an angle is ''derived'' from3 KB (575 words) - 12:41, 14 February 2011
- real finite-dimensional inner product space; possibly with translations defined on it.122 bytes (14 words) - 04:50, 2 September 2009
- A vector space that is endowed with an inner product and the corresponding norm.116 bytes (17 words) - 13:40, 4 January 2009
- ...uct]] of two vectors (resulting into a vector) and the dot indicates the [[inner product]] between two vectors (a scalar).4 KB (583 words) - 13:16, 25 November 2009
- linear operator on an inner product space that is equal to its Hermitian adjoint; also called self-adjoint oper152 bytes (21 words) - 08:31, 9 July 2009
- ...ear functional over a complex (real) Hilbert space can be expressed as the inner product of elements of that space with some fixed element of the space (this is kno [[Inner product space]]2 KB (258 words) - 12:33, 4 January 2009
- The notation ⟨ψ|φ⟩ for the inner product of ψ and φ, and related notations.137 bytes (20 words) - 13:59, 26 July 2008
- ...elates the absolute value of the [[inner product]] of two elements of an [[inner product space]] with the magnitude of the two said vectors. It is named in the hono ==The inequality for inner product spaces==4 KB (629 words) - 16:46, 17 December 2008
- Polynomials ''H''<sub>''n''</sub> that are orthogonal with respect to an inner product defined as <math>\left(H_{n'},H_n\right)\equiv\int_{-\infty}^{\infty} H_{n'241 bytes (43 words) - 09:11, 30 January 2009
- *[[Riemann manifold]] Differentiable [[manifold]] with non-degenerate [[inner product]].310 bytes (33 words) - 07:04, 7 February 2009
- {{r|Inner product}}347 bytes (48 words) - 14:08, 26 July 2008
- {{r|Inner product space}}359 bytes (48 words) - 15:04, 28 July 2009
- ...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 ℝ<sup>3</sup>×ℝ<sup>3</sup> into ℝ). The inner product satisfies9 KB (1,373 words) - 06:21, 11 December 2009
- {{r|Inner product space}}617 bytes (78 words) - 18:24, 11 January 2010
- {{r|Inner product}}576 bytes (77 words) - 19:04, 11 January 2010
- {{r|inner product space}}492 bytes (60 words) - 15:09, 28 July 2009
- {{r|Inner product space}}565 bytes (76 words) - 19:05, 11 January 2010
- ...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<2 KB (301 words) - 06:39, 21 October 2007
- This relation can be immediately generalized to ''m''-dimensional inner product spaces. Let the space ''V''<sub>''m''</sub> 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 comm8 KB (1,366 words) - 09:12, 10 August 2009
- {{r|Inner product}}942 bytes (125 words) - 18:29, 11 January 2010
- ...ite inner product between the elements ψ and φ of the same complex inner product space. We follow the physical convention11 KB (1,759 words) - 10:02, 2 August 2008
- ...[[eigenvector]]s, [[linear operators]], [[matrix representation]]s, and [[inner product]]s find broad application in [[science]].975 bytes (135 words) - 02:13, 3 September 2010
- {{r|Inner product}}885 bytes (142 words) - 15:23, 28 November 2008
- The distance is defined by means of the following positive definite [[inner product]] on ℝ<sup>''n''</sup>, ...b>i </sub> of '''y'''. Further, ⟨'''a''', '''b'''⟩ stands for an inner product between '''a''' and '''b'''.9 KB (1,403 words) - 02:22, 14 October 2013
- {{r|Inner product space}}905 bytes (145 words) - 15:27, 28 November 2008
- {{r|Inner product}}940 bytes (149 words) - 15:13, 28 July 2009
- {{r|Inner product space}}955 bytes (150 words) - 15:15, 28 July 2009
- Formally, the Euclidean plane is a 2-dimensional [[affine space]] with [[inner product]].1 KB (163 words) - 15:47, 25 November 2008
- ...que complements each of the other). However, if ''V'' is in addition an [[inner product]] space, then there is a unique ''orthogonal complement''1 KB (195 words) - 15:17, 12 December 2008
- * [[Inner product space]]982 bytes (148 words) - 07:17, 3 December 2007
- ...c bracket, <math>\langle\cdot|\cdot\rangle</math>, which is equal to the [[inner product]] between them.4 KB (690 words) - 12:51, 26 March 2011
- ...oduct is usually notated as a bra and ket, following [[Dirac]]. Thus, the inner product of Φ and Ψ is written as, ...expressed by stating that Φ and Φ′ are orthogonal (have zero inner product).8 KB (1,273 words) - 11:29, 9 July 2009
- {{r|Inner product}}1 KB (187 words) - 20:18, 11 January 2010
- it does not define an inner product for random variables, but only for random variables with mean 0 or, equival4 KB (694 words) - 17:27, 25 August 2013
- it does not define an inner product for random variables, but only for random variables with mean 0 or, equival4 KB (694 words) - 17:28, 25 August 2013
- The integrand (expression under the integral) is an [[inner product]] of the vector '''B''' and the vector d'''S'''. The latter vector has the1 KB (217 words) - 05:55, 11 July 2008
- ===Finite-dimensional inner product space=== ...erator and its trace can be given. These expressions can be generalized to inner product spaces of infinite dimension and are of great importance in [[quantum mecha12 KB (1,903 words) - 10:57, 2 February 2009
- ...a linear operator ''T'' on an arbitrary complex Hilbert space ''H'', with inner product ⟨ ⋅, ⋅ ⟩<sub>''H''</sub>, could be defined generally a ...onjugate of the complex number ''a'' appears is due to the property of the inner product on complex Hilbert space. The fact that the multiplication order of the ope5 KB (914 words) - 08:41, 17 October 2009
- ...Its elements have well-defined length, namely, the square root of the inner product of the vector with itself. The distance between any two points ''P'' and ' ...an affine space with difference space ''V'' on which a positive-definite inner product is defined. Then ''A'' is called a ''Euclidean space''. The distance betw15 KB (2,366 words) - 09:09, 4 April 2010
- ...annian metric <math> g = \{g_p\}_{p \in M} </math> is a set of symmetric [[inner product]]s5 KB (805 words) - 17:01, 28 November 2008
- ...e = "vertical-align: top"> <math>\mathbb{R}^3</math></font> that preserve inner product (are [[isometric]]) and preserve orientation (have unit [[determinant]]). ...ter 9.10. Here it is proved that a linear operator on a finite-dimensional inner product space is normal if and only if it has a complete set of orthonormal eigenve12 KB (1,865 words) - 02:49, 19 April 2010
- ...\vec{a}</math> and <math>\vec{b}</math> in <math>\mathbb{R}^3</math> the [[inner product]] is invariant, that is, and observe that the inner product becomes by virtue of the orthonormality of the basis vectors22 KB (3,624 words) - 01:45, 26 October 2013
- ...e interaction is switched on, the symmetry of the atom is lowered to the [[inner product group]], which is isomorphic to SO(3) and is the subgroup of SO(3) × The inner product group consists of the following elements21 KB (3,338 words) - 10:45, 11 June 2009
- ...space structure; it requires imposing the additional structure given an [[inner product]] on the space. Compartmentalizing mathematical information in this way ca ...ll also be defined on the vector space, with the dot then representing the inner product between two vectors.15 KB (2,506 words) - 05:16, 11 May 2011
- ..., namely, as an affine space whose difference space is a three-dimensional inner product space. ====[[Normed space|Normed]], [[Banach space|Banach]], [[Inner product space|inner product]], and [[Hilbert space|Hilbert]] spaces====28 KB (4,311 words) - 08:36, 14 October 2010
- ...ment d''S'' and d'''''S''''' is of length d''S''. The dot stands for the [[inner product]] between the magnetic induction '''B''' and d'''''S'''''. In order to obtain the electromotive force (EMF) one must integrate the inner product of electric force with path, '''F'''<sub>Lor</sub>⋅d'''l''', counter-9 KB (1,549 words) - 12:18, 11 June 2009
- ...'''k'''-space. We will show that divergence in '''r'''-space becomes an [[inner product]] in '''k'''-space and a curl becomes a [[cross product]]. Thus, we define11 KB (1,756 words) - 14:38, 12 April 2009
- ...esimal element and direction normal (perpendicular) to the element. The [[inner product]] <math>\scriptstyle \mathbf{J}\cdot d\mathbf{S}</math> is the current ''d3 KB (510 words) - 10:16, 16 July 2008
- ...up of transformations]] of two-dimensional complex vectors leaving their [[inner product]] fixed, and hence also the norm of a complex vector.<ref name= Mirman>7 KB (1,096 words) - 05:49, 17 October 2013
- ...as a real [[affine space]] whose difference space is a three-dimensional [[inner product space]]. For further details see [[Affine space#Euclidean space]] and [[spa ...s points, vectors and [[scalar product]] (called also [[dot product]] or [[inner product]]) of vectors without mentioning linear and affine spaces. Optionally, Cart16 KB (2,609 words) - 03:09, 8 March 2024
- ...as a real [[affine space]] whose difference space is a three-dimensional [[inner product space]]. For further details see [[Affine space#Euclidean space]] and [[spa ...s points, vectors and [[scalar product]] (called also [[dot product]] or [[inner product]]) of vectors without mentioning linear and affine spaces. Optionally, Cart16 KB (2,638 words) - 03:10, 8 March 2024
- ...(here <math> \scriptstyle \langle \cdot,\cdot\rangle </math> denotes the [[inner product]] between two vectors in the associated Hilbert space), and immediately aft5 KB (726 words) - 01:57, 30 September 2009
- ...''H''<sub>''n''</sub>(''x'') are orthogonal in the sense of the following inner product:4 KB (580 words) - 06:31, 31 May 2009
- ...egrable functions on <math>\scriptstyle \mathbb{R}</math> with the complex inner product <math>\scriptstyle \langle f,g\rangle=\int_{-\infty}^{\infty}f(x)\overline{4 KB (709 words) - 06:58, 23 December 2008
- Indeed, when we write the integral as an [[inner product]], it follows from partial integration and the vanishing of ''f''(''x'') o14 KB (2,220 words) - 12:28, 26 October 2021
- The surface integral contains the [[inner product]] between the current density '''J''' (a vector) and a vector d'''S''' ort6 KB (972 words) - 16:59, 27 October 2021
- ...uld be, in general, ambiguous. For instance, in classical mechanics the [[inner product]] '''p'''⋅'''r''' is equal to '''r'''⋅'''p''', but simple replace ...radient operator '''∇''' to the new coordinates and remember that an inner product involves, in general, a [[metric tensor]] '''g'''. Because the metric tenso13 KB (1,900 words) - 10:49, 30 November 2009
- ...''</sub> is the position of the ''i'' th particle; the dot stands for an [[inner product]] between the two 3-[[vector]]s. Indicate long-time averages by angular b7 KB (1,257 words) - 03:23, 24 March 2010
- ...ly, as an [[affine space]] whose difference space is a three-dimensional [[inner product space]]. For further details see [[Affine space#Euclidean space]] and [[spa10 KB (1,620 words) - 03:09, 8 March 2024
- ...ly, as an [[affine space]] whose difference space is a three-dimensional [[inner product space]]. For further details see [[Affine space#Euclidean space]] and [[spa10 KB (1,620 words) - 03:09, 8 March 2024
- ...any Cartesian frame centered anywhere in space, because distance, being an inner product, is invariant under rotation of the frame and, being the norm of a diffe ...artial^2}{\partial z_i^2} </math>. Since the kinetic energy operator is an inner product, it is invariant under rotation of the Cartesian frame with respect to whic31 KB (4,757 words) - 02:20, 27 October 2013
- The [[inner product]] between two state vectors is a complex number known as a [[probability am37 KB (5,578 words) - 04:54, 21 March 2024
- ...qual to the length ''a'' of the semi-major axis. For the following two [[inner product]]s (indicated by a centered dot) we find,23 KB (3,849 words) - 06:03, 29 August 2013
- ...qual to the length ''a'' of the semi-major axis. For the following two [[inner product]]s (indicated by a centered dot) we find,23 KB (3,852 words) - 20:27, 9 January 2021