Norm (mathematics) - Citizendia

In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector. Linear algebra is the branch of Mathematics concerned with For functional analysis as used in psychology see the Functional analysis (psychology article Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Linear algebra, the null vector or zero vector is the vector (0 0 &hellip 0 in Euclidean space, all of whose components are zero A seminorm (or pseudonorm), on the other hand, is allowed to assign zero length to some non-zero vectors.

A simple example is the 2-dimensional Euclidean space R² equipped with the Euclidean norm. Elements in this vector space (e. g. , (3, 7) ) are usually drawn as arrows in a 2-dimensional cartesian coordinate system starting at the origin (0, 0). In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane The Euclidean norm assigns to each vector the length of its arrow. Because of this, the Euclidean norm is often known as the magnitude. The magnitude of a mathematical object is its size a property by which it can be larger or smaller than other objects of the same kind in technical terms an Ordering

A vector space with a norm is called a normed vector space. In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to Similarly, a vector space with a seminorm is called a seminormed vector space. In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to

1 Definition
2 Examples
3 Properties
4 Absolutely convex and absorbing sets
5 Notes
6 References
7 See also

Definition

Given a vector space V over a subfield F of the complex numbers such as the complex numbers themselves or the real or rational numbers, a seminorm on V is a function $p:V\to\mathbb{R}; x\mapsto{}p(x)$ with the following properties:

For all a in F and all u and v in V,

p(a v) = |a| p(v), (positive homogeneity or positive scalability)
p(u + v) ≤ p(u) + p(v) (triangle inequality or subadditivity). In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater In Mathematics, subadditivity is a property of a function that states roughly that evaluating the function for the sum of two elements of the Domain

A simple consequence of these two axioms, positive homogeneity and the triangle inequality, is p(0) = 0 and thus

p(v) ≥ 0 (positivity).

A norm is a seminorm with the additional property

p(v) = 0 if and only if v is the zero vector (positive definiteness). In Linear algebra, the null vector or zero vector is the vector (0 0 &hellip 0 in Euclidean space, all of whose components are zero

A norm is usually denoted ||v||, and sometimes |v|, instead of p(v).

Although every vector space is seminormed (e. g. , with the trivial seminorm in the Examples section below), it may not be normed. Every vector space V with seminorm p(v) induces a normed space V/W, called the quotient space, where W is the subspace of V consisting of all vectors v in V with p(v) = 0. In Linear algebra, the quotient of a Vector space V by a subspace N is a vector space obtained by "collapsing" N The induced norm on V/W is given by ||W+v|| = p(v) and is clearly well-defined.

A topological vector space is called normable (seminormable) if the topology of the space can be induced by a norm (seminorm). In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of

Examples

All norms are seminorms.
The trivial seminorm, with p(x) = 0 for all x in V.
The absolute value is a norm on the real numbers. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. In Mathematics, the real numbers may be described informally in several different ways
Every linear form f on a vector space defines a seminorm by x → |f(x)|. This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional

Euclidean norm

On Rⁿ, the intuitive notion of length of the vector x = [x₁, x₂, . . . , x_n] is captured by the formula

$\|\mathbf{x}\| := \sqrt{x_1^2 + \cdots + x_n^2}.$

This gives the ordinary distance from the origin to the point x, a consequence of the Pythagorean theorem. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry The Euclidean norm is by far the most commonly used norm on Rⁿ, but there are other norms on this vector space as will be shown below.

On Cⁿ the most common norm is

$\|\mathbf{z}\| := \sqrt{|z_1|^2 + \cdots + |z_n|^2}.$ , equivalent with the Euclidean norm on R²ⁿ.

In each case we can also express the norm as the square root of the inner product of the vector and itself. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. The Euclidean norm is also called the l ², see L^p space. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding

The set of vectors whose Euclidean norm is a given constant forms the surface of a sphere. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

Taxicab norm or Manhattan norm

Main article: Taxicab geometry

$\|\emph{\textbf{x}}\|_1 := \sum_{i=1}^{n} |x_i|.$

The name relates to the distance a taxi has to drive in a rectangular street grid to get from the origin to the point x. Taxicab geometry, considered by Hermann Minkowski in the 19th century is a form of Geometry in which the usual metric of Euclidean geometry The grid plan or gridiron plan is a type of City plan in which Streets run at right angles to each other forming a grid.

The set of vectors whose 1-norm is a given constant forms the surface of a cross polytope. In Geometry, a cross-polytope, or orthoplex, or hyperoctahedron, is a regular, convex Polytope that exists in any number of dimensions

p-norm

Let p≥1 be a real number.

$\|\emph{\textbf{x}}\|_p := \left( \sum_{i=1}^n |x_i|^p \right)^\frac{1}{p}.$

Note that for p = 1 we get the taxicab norm and for p = 2 we get the Euclidean norm. See also L^p space. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding

Infinity norm or maximum norm

Main article: Maximum norm

$\|\emph{\textbf{x}}\|_\infty := \max \left(|x_1|, \ldots ,|x_n| \right).$

The set of vectors whose ∞-norm is a given constant forms the surface of a hypercube. In Mathematical analysis, the uniform norm assigns to real- or complex -valued bounded functions f the nonnegative number In Geometry, a hypercube is an n -dimensional analogue of a square ( n = 2 and a Cube ( n = 3

Zero norm

In the machine learning and optimization literature, one often finds reference to the zero norm. Machine learning is a subfield of Artificial intelligence that is concerned with the design and development of Algorithms and techniques that allow computers to "learn" In Mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function The zero norm of x is defined as $\lim_{p\rightarrow 0} \|x\|_p^p,$ where $\|x\|_p$ is the p-norm defined above. If we define $0^0 \ \stackrel{\mathrm{def}}{=}\ 0$ then we can write the zero norm as $\sum_{i=1}^n x_i^0$ . In Mathematics, defined and undefined are used to explain whether or not expressions have meaningful sensible and unambiguous values It follows that the zero norm of x is simply the number of non-zero elements of x. Despite its name, the zero norm is not a true norm; in particular, it is not positive homogeneous. Such a norm can be defined over arbitrary fields (besides the fields of complex numbers). In the context of the information theory, it is often called the Hamming distance in the case of the 2-element GF(2) field. Information theory is a branch of Applied mathematics and Electrical engineering involving the quantification of Information. Examples The Hamming distance between 1011101 and 1001001 In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements

Other norms

Other norms on Rⁿ can be constructed by combining the above; for example

$\|\emph{\textbf{x}}\| := 2|x_1| + \sqrt{3|x_2|^2 + \max(|x_3|,2|x_4|)^2}$

is a norm on R⁴.

For any norm and any bijective linear transformation A we can define a new norm of x, equal to

$\|Ax\|.$

In 2D, with A a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In 2D, each A applied to the taxicab norm, up to inversion and interchanging of axes, gives a different unit ball: a parallelogram of a particular shape, size and orientation. In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides In 3D this is similar but different for the 1-norm (octahedrons) and the maximum norm (prisms with parallelogram base). An octahedron (plural octahedra is a Polyhedron with eight faces General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces

All the above formulas also yield norms on Cⁿ without modification.

Infinite dimensional case

The generalization of the above norms to an infinite number of components leads to the L^p spaces, with norms

$\|x\|_p = \left(\sum_{i\in\mathbb N}|x_i|^p\right)^{\frac1p}$ resp. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding $\|f\|_{p,X} = \left(\int_X|f(x)|^p\,\mathrm dx\right)^{\frac1p}$

(for complex-valued sequences x resp. functions f defined on $X\subset\mathbb R$ ), which can be further generalized (see Haar measure). In Mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of Locally compact topological groups and subsequently define

Any inner product induces in a natural way the norm $\|x\| := \sqrt{\langle x,x\rangle}.$

Other examples of infinite dimensional normed vector spaces can be found in the Banach space article. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis

Properties

Illustrations of unit circles in different norms. In Mathematics, a unit circle is

The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm the unit circle in R² is a rhomboid, for the 2-norm (Euclidean norm) it is the well-known unit circle, while for the infinity norm it is a square. In Mathematics, a unit circle is This article is about mathematics For Rhomboid muscles in anatomy see Rhomboid major muscle and Rhomboid minor muscle. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides For any p-norm it is a superellipse (with congruent axes). The superellipse (or Lamé curve) is the geometric figure defined in the Cartesian coordinate system as the set of all points ( x, y) with See the accompanying illustration.

In terms of the vector space, the seminorm defines a topology on the space, and this is a Hausdorff topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space

Two norms ||•||_α and ||•||_β on a vector space V are called equivalent if there exist positive real numbers C and D such that

$C\|x\|_\alpha\leq\|x\|_\beta\leq D\|x\|_\alpha$

for all x in V. On a finite-dimensional vector space all norms are equivalent. For instance, the $l 1$ , $l 2$ , and $l_\infty$ norms are all equivalent on $\mathbb{R}^n$ :

$\|x\|_2\le\|x\|_1\le\sqrt{n}\|x\|_2$

$\|x\|_\infty\le\|x\|_2\le\sqrt{n}\|x\|_\infty$

$\|x\|_\infty\le\|x\|_1\le n\|x\|_\infty$

Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished. To be more precise the uniform structure defined by equivalent norms on the vector space is uniformly isomorphic. In the mathematical field of Topology a uniform isomorphism or uniform homeomorphism is a special Isomorphism between Uniform spaces

Every (semi)-norm is a sublinear function, which implies that every norm is a convex function. A sublinear function in Linear algebra and related areas of Mathematics, is a function f V \rightarrow \mathbf{F} on a vector space V over In Mathematics, a real-valued function f defined on an interval (or on any Convex subset of some Vector space) is called convex As a result, finding a global optimum of a norm-based objective function is often tractable. In Mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function

Given a finite family of seminorms p_i on a vector space the sum

$p(x):=\sum_{i=0}^n p_i(x)$

is again a seminorm.

For any norm p on a vector space V, we have that for all u and v ∈ V:

p(u ± v) ≥ | p(u) − p(v) |

For the l^p norms, we have^[1]

$|x^\top y|\le\| x\|_p\|y\|_q\qquad \frac{1}{p}+\frac{1}{q}=1$

A special case of the above property is the Cauchy-Schwarz inequality:^[1]

$|x^\top y|\le\|x\|_2\|y\|_2$

Absolutely convex and absorbing sets

Seminorms are closely related to absolutely convex and absorbing sets. In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding In Mathematics, the Cauchy–Schwarz inequality, also known as the Schwarz inequality, the Cauchy inequality, or the Cauchy–Schwarz–Bunyakovsky A set C in a real or complex Vector space is said to be absolutely convex if it is convex and balanced. In Functional analysis and related areas of Mathematics an absorbing set in a Vector space is a set S which can be inflated Let p be a seminorm on a vector space V, then for any scalar α the sets {x : p(x) < α} and {x : p(x) ≤ α} are absorbing and absolutely convex. In a normed vector space the set {x : p(x) ≤ 1} is called the closed unit ball. In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed

Conversely to each absorbing and absolutely convex subset A of V corresponds a seminorm p called the gauge of A, defined as

p(x) := inf{α : α > 0, x ∈ α A}

with the property that

{x : p(x) < 1} ⊆ A ⊆ {x : p(x) ≤ 1}. In Functional analysis, given a linear space X, a Minkowski functional is a device that uses the linear structure to introduce a topology on X. In Mathematics the infimum of a Subset of some set is the Greatest element, not necessarily in the subset that is less than or equal to all elements of

A locally convex topological vector space has a local basis consisting of absolutely convex and absorbing sets. In Functional analysis and related areas of Mathematics, locally convex topological vector spaces or locally convex spaces are examples of Topological In Topology and related areas of Mathematics, the neighbourhood system or neighbourhood filter \mathcal{V}(x for a point x is the A common method to construct such a basis is to use a family of seminorms. Typically this family is infinite, and there are enough seminorms to distinguish between elements of the vector space, creating a Hausdorff space.

Notes

^ ^a ^b Golub, Gene; Charles F. Gene Howard Golub ( February 29, 1932 &ndash November 16, 2007) Fletcher Jones Professor of Computer Science (and by courtesy of Electrical Van Loan (1996). Matrix Computations - Third Edition. Baltimore: The Johns Hopkins University Press, 53. ISBN 0-8018-5413-X.

References

Bourbaki, N. (1987). Topological Vector Spaces, Chapters 1-5, Elements of Mathematics. Springer. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic ISBN 3-540-13627-4.

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