Векторлық кеңістіктердегі нормалар мен өлшемдер

Length in a vector space
norms of normed vector spaces
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 from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2 norm, or, sometimes, the magnitude of the vector. This norm can be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "" in the homogeneity axiom. It can also refer to a norm that can take infinite values, or to certain functions parametrised by a directed set.

The Euclidean norm of a complex number is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean plane This identification of the complex number as a vector in the Euclidean plane, makes the quantity (as first suggested by Euler) the Euclidean norm associated with the complex number. For , the norm can also be written as where is the complex conjugate of

In metric geometry, the discrete metric takes the value one for distinct points and zero otherwise. When applied coordinate wise to the elements of a vector space, the discrete distance defines the Hamming distance, which is important in coding and information theory. In the field of real or complex numbers, the distance of the discrete metric from zero is not homogeneous in the non zero point; indeed, the distance from zero remains one as its non zero argument approaches zero. However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness. When applied component wise to vectors, the discrete distance from zero behaves like a non homogeneous "norm", which counts the number of non zero components in its vector argument; again, this non homogeneous "norm" is discontinuous. In signal processing and statistics, David Donoho referred to the zero "norm" with quotation marks. Following Donoho's notation, the zero "norm" of is simply the number of non zero coordinates of or the Hamming distance of the vector from zero. When this "norm" is localized to a bounded set, it is the limit of norms as approaches 0. Of course, the zero "norm" is not truly a norm, because it is not positive homogeneous. Indeed, it is not even an F norm in the sense described above, since it is discontinuous, jointly and severally, with respect to the scalar argument in scalar–vector multiplication and with respect to its vector argument. Abusing terminology, some engineers omit Donoho's quotation marks and inappropriately call the number of non zeros function the norm, echoing the notation for the Lebesgue space of measurable functions.

Let be a finite extension of a field of inseparable degree and let have algebraic closure If the distinct embeddings of are then the Galois theoretic norm of an element is the value As that function is homogeneous of degree , the Galois theoretic norm is not a norm in the sense of this article. However, the th root of the norm (assuming that concept makes sense) is a norm.

The concept of norm in composition algebras does not share the usual properties of a norm since null vectors are allowed. A composition algebra consists of an algebra over a field an involution and a quadratic form called the "norm". The characteristic feature of composition algebras is the homomorphism property of : for the product of two elements and of the composition algebra, its norm satisfies In the case of division algebras and the composition algebra norm is the square of the norm discussed above. In those cases the norm is a definite quadratic form. In the split algebras the norm is an isotropic quadratic form.