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...

analysis, a Banach space (/ˈbɑː.nʌx/, Polish pronunciation: [ˈba.nax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric...

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...

associated norm, (denoted | x | {\displaystyle |x|} and | y | {\displaystyle |y|} in the picture); so, every inner product space is a normed vector space. If...

property characterizes strictly convex normed spaces such as the ℓp spaces with 1 < p < ∞. However, there are normed spaces in which this is not true. For instance...

a norm defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm ‖ T ‖ {\displaystyle \|T\|}...

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...

Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces (normed vector spaces that...

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...

a norm, such as the Euclidean space, is called a normed vector space. The norm of a vector v in a normed vector space can be considered to be the magnitude...

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...

{\displaystyle L^{\infty }} norm has the highest equilateral dimension among all normed spaces. Petty (1971) asked whether every normed vector space of dimension d...

commonly used for the initial topology of a topological vector space (such as a normed vector space) with respect to its continuous dual. The remainder of...

{\displaystyle V} is a normed vector space (for example, a Banach space or a Hilbert space) then the strong topology on V ′ {\displaystyle V'} is normed (in fact a...

normed vector space (X, || ||) is strictly convex if and only if x ≠ y and || x || = || y || = 1 together imply that || x + y || < 2. A normed vector...

mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes...

. {\displaystyle p\in X.} Any normed vector space V with norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is also a metric space with the metric d ( x , y ) = ‖...

topological vector spaces, which include function spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces. In this article, vectors are...

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...

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...

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 partial...

and therefore admit the structure of a metric space, including Riemannian manifolds, normed vector spaces, and graphs. In abstract algebra, the p-adic...

normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, where the norm arises...

of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis, when a function f :...

sufficient continuous linear functionals defined on every normed vector space in order to study the dual space. Another version of the Hahn–Banach theorem is known...

topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be...

In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous...

components. When the underlying vector space X {\displaystyle X} is a (not necessarily finite-dimensional) normed vector space, analytic questions, irrelevant...

span of a set of vectors is the minimal closed set which contains the linear span of that set. Suppose that X is a normed vector space and let E be any...

norm – Norm on a vector space of matrices Minkowski functional – Function made from a set Norm (mathematics) – Length in a vector space Normed vector...