In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance...
34 KB (5,699 words) - 02:44, 6 September 2024
mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. A norm is a generalization...
18 KB (2,890 words) - 22:11, 21 February 2024
of mathematics, norms are defined for elements within a vector space. Specifically, when the vector space comprises matrices, such norms are referred to...
26 KB (4,513 words) - 19:16, 28 August 2024
Banach space (redirect from Complete normed vector space)
normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and...
103 KB (17,213 words) - 23:33, 24 August 2024
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase...
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space of bounded linear operators between two given normed vector spaces. Informally, the operator norm ‖ T ‖ {\displaystyle \|T\|} of a linear map T : X...
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Inner product space (redirect from Orthogonal vector)
|y|} in the picture); so, every inner product space is a normed vector space. If this normed space is also complete (that is, a Banach space) then the...
56 KB (7,305 words) - 10:53, 26 August 2024
topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces. In this article, vectors are...
87 KB (11,494 words) - 16:23, 2 August 2024
Ball (mathematics) (section In normed vector spaces)
{\displaystyle B_{1}[p]=X} for any p ∈ X . {\displaystyle p\in X.} Any normed vector space V with norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is also a metric space with...
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the dual norm is a measure of size for a continuous linear function defined on a normed vector space. Let X {\displaystyle X} be a normed vector space with...
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Dot product (redirect from Norm squared)
space is a normed vector space, and the inner product of a vector with itself is real and positive-definite. The dot product is defined for vectors that have...
28 KB (4,320 words) - 10:11, 28 July 2024
v by k. A vector space equipped with a norm is called a normed vector space (or normed linear space). The norm is usually defined to be an element of...
8 KB (1,044 words) - 01:38, 21 June 2024
Magnitude (mathematics) (section Normed vector spaces)
applied as the measure of units between a number and zero. In vector spaces, the Euclidean norm is a measure of magnitude used to define a distance between...
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a length or size to any vector in a vector space Matrix norm, a map that assigns a length or size to a matrix Operator norm, a map that assigns a length...
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structure of gradation Normed vector space, a vector space on which a norm is defined Hilbert space Ordered vector space, a vector space equipped with a...
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Triangle inequality (section Normed vector space)
other geometries, the triangle inequality is a theorem about vectors and vector lengths (norms): ‖ u + v ‖ ≤ ‖ u ‖ + ‖ v ‖ , {\displaystyle \|\mathbf {u}...
34 KB (5,176 words) - 13:40, 1 July 2024
Fréchet space (redirect from Countably-normed space)
topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the metric induced by the norm). All...
29 KB (5,027 words) - 09:06, 22 August 2024
Vector notation In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more...
24 KB (3,265 words) - 03:31, 6 September 2024
Every normed vector space has a natural topological structure: the norm induces a metric and the metric induces a topology. This is a topological vector space...
103 KB (13,529 words) - 03:12, 5 July 2024
physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude...
61 KB (9,095 words) - 04:40, 8 September 2024
complex normed vector spaces do not have inner products, but all normed vector spaces have norms (by definition). For example, a commonly used norm for a...
9 KB (1,630 words) - 20:56, 31 August 2024
Dual space (redirect from Norm dual)
In mathematics, any vector space V {\displaystyle V} has a corresponding dual vector space (or just dual space for short) consisting of all linear forms...
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convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can...
58 KB (10,592 words) - 23:16, 10 August 2024
Bounded operator (section In normed vector spaces)
{\displaystyle Y.} If X {\displaystyle X} and Y {\displaystyle Y} are normed vector spaces (a special type of TVS), then L {\displaystyle L} is bounded...
15 KB (2,471 words) - 12:14, 16 July 2024
q ‖ {\displaystyle \left\|\mathbf {q} \right\|} is the norm of vector q. The norm of a vector is calculated as such: ‖ q ‖ = ∑ i = 1 n q i 2 {\displaystyle...
10 KB (1,414 words) - 05:24, 24 June 2024
Quaternion (redirect from Quaternion norm)
and its vector part: q = q s + q → v . {\displaystyle q=q_{s}+{\vec {q}}_{v}.} Decompose the vector part further as the product of its norm and its versor:...
96 KB (12,653 words) - 17:28, 27 August 2024
Functional analysis (section Normed vector spaces)
is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear...
20 KB (2,499 words) - 18:10, 28 July 2024
Metric space (redirect from Norm induced metric)
admit the structure of a metric space, including Riemannian manifolds, normed vector spaces, and graphs. In abstract algebra, the p-adic numbers arise as...
79 KB (11,080 words) - 19:10, 3 September 2024