product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more). Algebraically... 27 KB (4,274 words) - 01:14, 27 February 2024 |
indefinite inner product space ( K , ⟨ ⋅ , ⋅ ⟩ , J ) {\displaystyle (K,\langle \cdot ,\,\cdot \rangle ,J)} is an infinite-dimensional complex vector space K {\displaystyle... 11 KB (1,953 words) - 13:08, 16 May 2022 |
Orthogonal complement (redirect from Annihilating space) vector space equipped with the usual dot product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } (thus making it an inner product space), and let... 13 KB (2,055 words) - 15:22, 7 May 2024 |
semidirect products Product of rings Ideal operations, for product of ideals Scalar multiplication Matrix multiplication Inner product, on an inner product space... 2 KB (246 words) - 18:54, 26 February 2024 |
Square-integrable function (redirect from L2-inner product) specifically a Hilbert space, because the space is complete under the metric induced by the inner product. This inner product space is conventionally denoted... 6 KB (888 words) - 19:40, 16 January 2024 |
mathematics, particularly linear algebra, an orthonormal basis for an inner product space V {\displaystyle V} with finite dimension is a basis for V {\displaystyle... 14 KB (2,582 words) - 15:17, 7 May 2024 |
tensor product. The tensor product allows Hilbert spaces to be collected into a symmetric monoidal category. Since Hilbert spaces have inner products, one... 12 KB (1,996 words) - 20:53, 17 April 2024 |
complete for this norm. An inner product space is a normed vector space whose norm is the square root of the inner product of a vector and itself. The... 18 KB (2,901 words) - 22:11, 21 February 2024 |
inner product for Euclidean vector spaces, better known as the dot product. The dot product is the trace of the outer product. Unlike the dot product... 18 KB (2,945 words) - 11:03, 23 February 2024 |
Mirror (2022) Priam InnerSpace, a hard disk drive series by Priam Corporation in the 1980s Inner product space, a kind of vector space in linear algebra... 1 KB (191 words) - 19:56, 8 March 2024 |
Pythagorean theorem (section Inner product spaces) with infinitely many components in an inner product space, as in functional analysis. In an inner product space, the concept of perpendicularity is replaced... 92 KB (12,566 words) - 21:51, 4 May 2024 |
defined over a set of labels S {\displaystyle S} in an inner product space with an inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle... 3 KB (379 words) - 07:15, 19 August 2023 |
Cauchy–Schwarz inequality (category Pages displaying short descriptions with no spaces via Module:Annotated link) inequality) is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered... 37 KB (5,154 words) - 11:04, 11 May 2024 |
vectors defined in an inner product space. Cosine similarity is the cosine of the angle between the vectors; that is, it is the dot product of the vectors divided... 21 KB (3,005 words) - 19:29, 15 February 2024 |
of an inner product. Physical space can be modelled as a vector space which additionally has the structure of an inner product. The inner product defines... 34 KB (4,829 words) - 01:43, 17 April 2024 |
Functional (mathematics) (section Inner product spaces) . {\displaystyle \{{\vec {x}}\}^{\perp }.} For example, taking the inner product with a fixed function g ∈ L 2 ( [ − π , π ] ) {\displaystyle g\in L^{2}([-\pi... 9 KB (1,444 words) - 15:39, 16 November 2023 |
Bra–ket notation (redirect from Bra-ket notation for outer product) an inner product ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} with antilinear first argument, which makes V {\displaystyle V} an inner product space. Then... 43 KB (6,393 words) - 23:40, 12 May 2024 |
translations which is equipped with an inner product. The action of translations makes the space an affine space, and this allows defining lines, planes... 47 KB (6,957 words) - 21:59, 2 May 2024 |
function space. The inner product space is then called complete. A complete inner product space is a Hilbert space. The abstract state space is always... 99 KB (13,534 words) - 13:10, 15 April 2024 |
strictly positive. A semi-inner-product, L-semi-inner product, or a semi-inner product in the sense of Lumer for a linear vector space V {\displaystyle V} over... 7 KB (1,184 words) - 06:10, 2 February 2024 |
Ptolemy's inequality (section Inner product spaces) generally in any inner product space, and whenever it is true for a real normed vector space, that space must be an inner product space. For other types... 11 KB (1,433 words) - 18:52, 9 November 2023 |
well as physics, a linear operator A {\displaystyle A} acting on an inner product space is called positive-semidefinite (or non-negative) if, for every x... 6 KB (1,100 words) - 23:28, 3 January 2024 |
Pons asinorum (section In inner product spaces) The isosceles triangle theorem holds in inner product spaces over the real or complex numbers. In such spaces, given vectors x, y, and z, the theorem... 16 KB (2,091 words) - 07:05, 2 May 2024 |
Kernel (linear algebra) (redirect from Left null space) {Nullity} (L)=\dim \left(\operatorname {domain} L\right).} When V is an inner product space, the quotient V / ker ( L ) {\displaystyle V/\ker(L)} can be identified... 23 KB (3,702 words) - 20:31, 30 April 2024 |
{\displaystyle \varphi :V\times V\to K} In this case V is called an inner product space. For example, if K is the field of real or complex numbers, any positive... 50 KB (6,281 words) - 00:00, 5 May 2024 |