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In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. A semi-norm on the other hand is allowed to assign zero length to some non-zero vectors. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...
Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...
Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...
In mathematics, a function returns a unique output for a given input. ...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
In linear algebra and related areas of mathematics, the null vector or zero vector in a vector space is the uniquely-determined vector, usually written 0, that is the identity element for vector addition. ...
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). The Euclidean norm assigns to each vector the length of its arrow. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...
A vector space with a norm is called a normed vector space. Similarly, a vector space with a semi-norm is called a semi-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 be easily extended to any real vector space Rn. ...
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...
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 semi-norm on V is a function p:V→R; x→ p(x) with the following properties: A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of minus one (−1), which cannot be represented by any real number. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In mathematics, a function returns a unique output for a given input. ...
For all a in F and all u and v in V,
- p(v) ≥ 0 (positivity)
- p(a v) = |a| p(v), (positive homogeneity or positive scalability)
- p(u + v) ≤ p(u) + p(v) (triangle inequality or subadditivity).
A norm is a semi-norm with the additional property In mathematics, the triangle inequality states that the distance from A to B to C is never shorter than going directly from A to C. The triangle inequality is a theorem in spaces such as the real numbers, Euclidean space, Lp spaces (p ≥ 1) and in all inner product spaces...
A sequence { an }, n ≥ 1, is called subadditive if it satisfies the inequality for all m and n. ...
- p(v) = 0 if and only if v is the zero vector (positive definiteness)
A topological vector space is called normable (semi-normable) if the topology of the space can be induced by a norm (semi-norm). In linear algebra and related areas of mathematics, the null vector or zero vector in a vector space is the uniquely-determined vector, usually written 0, that is the identity element for vector addition. ...
In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard...
Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
Notes Semi-norms are often denoted by p(v) (function notation) whereas norms are traditionally denoted ||v|| (as a variant of the absolute-value notation).
A useful consequence of the norm axioms is the inequality
- ||u ± v|| ≥ | ||u|| − ||v|| |
for all u and v ∈ K.
Examples - All norms are semi-norms
- The trivial semi-norms, those where p(x) = 0 for all x in V.
- The absolute value is a norm on the real numbers.
- Every linear form f on a vector space defines a semi-norm by x→|f(x)|.
In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...
Please refer to Real vs. ...
In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ...
Euclidean norm On Rn, the intuitive notion of length of the vector x = [x1, x2, ..., xn] is captured by the formula  This gives the ordinary distance from the origin to the point x, a consequence of the Pythagorean theorem. 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. The Pythagorean theorem: The sum of the areas of the two squares on the legs equals the area of the square on the hypotenuse. ...
On Cn the most common norm is
 , 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. // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity α resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ...
Taxicab norm or Manhattan norm Main article Taxicab geometry Manhattan distance sample: the three routes (red, blue, yellow) between the two points have the same length Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual metric of Euclidean geometry is replaced by a new metric in which the distance...
 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. The grid plan is a type of city plan in which streets run at right angles to each other, forming a grid. ...
p-norm Let p≥1 be a real number.  Note that for p=1 we get the taxicab norm and for p=2 we get the Euclidean norm. See also Lp space. In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...
Infinity norm or maximum norm Main article maximum norm In mathematical analysis, the uniform norm assigns to real- or complex-valued functions f the nonnegative number This norm is also called the supremum norm or the Chebyshev norm. ...
Zero norm In the machine learning and optimization literature, one often finds reference to the zero norm. The zero norm of x is defined as  where  is the p-norm defined above. If we define  then we can write the zero norm as  . 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. ...
In mathematics, the term optimization refers to the study of problems that have the form Given: a function f : A R from some set A to the real numbers Sought: an element x0 in A such that f(x0) ≤ f(x) for all x in A (minimization) or such that...
In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible output. ...
Other norms Other norms on Rn can be constructed by combining the above; for example  is a norm on R4.
For any norm and any bijective linear transformation A we can define a new norm of x, equal to  . In 2D, with A a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. 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 3D this is similar but different for the 1-norm (octahedrons) and the maximum norm (prisms with parallelogram base). A parallelogram. ...
An octahedron (plural: octahedra) is a polyhedron with eight faces. ...
In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ...
All the above formulas also yield norms on Cn without modification.
Examples of infinite dimensional normed vector spaces can be found in the Banach space article. In addition, inner product space becomes a normed vector space if we define the norm as In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...
In mathematics, an inner product space is a vector space with additional structure, an inner product (also called scalar product or dot product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...
Properties 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. See the accompanying illustration. An illustration of vector norms File links The following pages link to this file: Norm (mathematics) Categories: GFDL images ...
Illustration of a unit circle. ...
Illustration of a unit circle. ...
This article is about mathematics. ...
A circle, in Euclidean geometry, is the set of all points at a fixed distance, called the radius, from a fixed point, the centre. ...
In plane geometry, a square is a polygon with four equal sides and equal angles. ...
In terms of the vector space, the semi-norm defines a topology on the space, and this is a Hausdorff topology precisely when the semi-norm can distinguish between distinct vectors, which is again equivalent to the semi-norm being a norm. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
Two norms ||·||1 and ||·||2 on a vector space V are called equivalent if there exist positive real numbers C and D such that
for all x in V. On a finite dimensional vector space all norms are equivalent.
Equivalent norms define the same notions of continuity and convergence and do not need to be distinguished for most purposes. 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 which respects uniform properties. ...
Every (semi)-norm is a sublinear function, which implies that every norm is a convex function. As a result, finding a global optimum of a norm-based objective function is often tractable. In linear algebra and related areas of mathematics a function on a vector space V over the field F (which is either the real numbers R or the complex numbers C) is called sublinear if for all scalars γ and vectors x and y (positive homogenity) (subadditivity) Examples Every (semi...
A convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Convex function on interval I.e. ...
Optimization is a branch of mathematics which is concerned with finding maxima and minima of real-valued functions. ...
Given a finite family of semi-norms pi on a vector space the sum
 is again a semi-norm.
Absolutely convex and absorbing sets Semi-norms are closely related to absolutely convex and absorbing sets. Let p be a semi-norm 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 unit ball. In mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. ...
In functional analysis and related areas of mathematics an absorbing set in a vector space is a set S which can be inflated to include any element of the vector space. ...
some unit spheres 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. ...
Conversely to each absorbing and absolutely convex subset A of V corresponds a semi-norm p called the gauge of A, defined as
- p(x) := inf{α : α > 0, x ∈ α A}
with the property that In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset, that is smaller than all other elements of the subset. ...
- {x : p(x) < 1} ⊆ A ⊆ {x : p(x) ≤ 1}.
A locally convex topological vector space has a local basis consisting of absolutely convex and absorbing sets. A common method to construct such a basis is to use a familiy of semi-norms. Typically this family is infinite, and there are enough semi-norms to distinguish between elements of the vector space, creating a Hausdorff space. In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoods of 0. ...
In topology and related areas of mathematics, the neighbourhood system or neighbourhood filter for a point x is the collection of all neighbourhoods for the point x. ...
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