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Encyclopedia > 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 any real vector space Rn. It turns out that the following properties of "vector length" are the crucial ones. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ...

  1. The zero vector, 0, has zero length; every other vector has a positive length.
  2. Multiplying a vector by a positive number has the same effect on the length.
  3. The triangle inequality holds. That is, taking norms as distances, the distance from A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line.

Their generalization for more abstract vector spaces, leads to the notion of norm. A vector space on which a norm is defined is then called a normed vector space. In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... 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. ...

Contents

Definition

A semi normed vector space is a pair (V,p) where V is a vector space and p a semi norm on V. In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... 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 normed vector space is a pair (V,||·||) where V is a vector space and ||·|| a norm on V. In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... 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. ...


We often omit p or ||·|| and just write V for a space if it is clear from the context what (semi) norm we are using.


Topological structure

If (V, ||·||) is a normed vector space, the norm ||·|| induces a notion of distance and therefore a topology on V. This distance is defined in the natural way: the distance between two vectors u and v is given by ||u-v||. This topology is precisely the weakest topology that makes ||·|| continuous. Furthermore, this natural topology is compatible with the linear structure of V in the following sense: A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ...

  1. The vector addition + : V × VV is jointly continuous with respect to this topology. This follows directly from the triangle inequality.
  2. The scalar multiplication · : K × VV, where K is the underlying scalar field of V, is jointly continuous. This follows from the triangle inequality and homogeneity of the norm.

Similarly, for any semi-normed vector space we can define the distance between two vectors u and v as ||u-v||. This turns the semi normed space into a semi metric space (notice this is weaker than a metric) and allows the definition of notions such as continuity and convergence. To put it more abstractly every semi normed vector space is a topological vector space and thus carries a topological structure which is induced by the semi-norm. In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ... In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ... In mathematics a topological vector space is one of the basic structures investigated in functional analysis. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...


Of special interest are complete normed spaces called Banach spaces. Every normed vector space V sits as a dense subspace inside a Banach space; this Banach space is essentially uniquely defined by V and is called the completion of V. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...


All norms on a finite-dimensional vector space are equivalent from a topological point as they induce the same topology (although the resulting metric spaces need not be the same). And since any Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are Banach spaces. A normed vector space V is finite-dimensional if and only if the unit ball B = {x : ||x|| ≤ 1} is compact, which is the case if and only if V is locally compact; this is a consequence of Riesz's lemma. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... Rieszs lemma is an lemma in functional analysis. ...


The topology of a semi normed vector has many nice properties. Given a neighbourhood system mathcal{N}(0) around 0 we can construct all other neighbourhood systems as 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. ...

mathcal{N}(x)= x + mathcal{N}(0) := {x + N mid N in mathcal{N}(0) }

with

x + N := {x + n mid n in N }.

Moreover there exists a neighbourhood basis for 0 consisting of absorbing and convex sets. As this property is very useful in functional analysis, generalizations of normed vector spaces with this property are studied under the name locally convex spaces. 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. ... 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. ... Look up Convex set in Wiktionary, the free dictionary. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In functional analysis and related areas of mathematics locally convex topological vector spaces or locally convex spaces are generalizations of semi normed spaces. ...


Linear maps and dual spaces

The most important maps between two normed vector spaces are the continuous linear maps. Together with these maps, normed vector spaces form a category. In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...


The norm is a continuous function on its vector space. All linear maps between finite dimensional vector spaces are also continuous.


An isometry between two normed vector spaces is a linear map f which preserves the norm (meaning ||f(v)|| = ||v|| for all vectors v). Isometries are always continuous and injective. A surjective isometry between the normed vector spaces V and W is called a isometric isomorphism, and V and W are called isometrically isomorphic. Isometrically isomorphic normed vector spaces are identical for all practical purposes. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...


When speaking of normed vector spaces, we augment the notion of dual space to take the norm into account. The dual V ' of a normed vector space V is the space of all continuous linear maps from V to the base field (the complexes or the reals) — such linear maps are called "functionals". The norm of a functional φ is defined as the supremum of |φ(v)| where v ranges over all unit vectors (i.e. vectors of norm 1) in V. This turns V ' into a normed vector space. An important theorem about continuous linear functionals on normed vector spaces is the Hahn-Banach theorem. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ... In mathematics, the supremum of an ordered set S is the least element that is greater than or equal to each element of S. Consequently, it is also referred to as the least upper bound (also lub and LUB). ... In mathematics, the Hahn-Banach theorem is a central tool in functional analysis. ...


Normed spaces as quotient spaces of semi normed spaces

The definition of many normed spaces (in particular, Banach spaces) involves a seminorm defined on a vector space and then the normed space is defined as the quotient space by the subspace of elements of seminorm zero. For instance, with the Lp spaces, the function defined by In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ... In mathematics, the Lp and lp spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

|f|_p = left( int |f(x)|^p ;dx right)^{1/p}

is a seminorm on the vector space of all functions on which the Lebesgue integral on the right hand side is defined and finite. However, the seminorm is equal to zero for any function supported on a set of Lebesgue measure zero. These functions form a subspace which we "quotient out", making them equivalent to the zero function. In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ... The word support has several specialized meanings: In mathematics, see support (mathematics). ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ...


Finite product spaces

Given n semi normed spaces Xi with semi norms pi we can define the product space as In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ...

X := prod_{i=1}^{n} X_i

with vector addition defined as

(x_1,ldots,x_n)+(y_1,ldots,y_n):=(x_1 + y_1, ldots x_n + y_n)

and scalar multiplication defined as

alpha(x_1,ldots,x_n):=(alpha x_1, ldots, alpha x_n).

We define a new function p

p:X mapsto mathbb{R}

as

p:(x_1,ldots,x_n) to sum_{i=1}^n p_i(x_i).

which is a semi norm on X. The function p is a norm if and only if all pi are norms.


Moreover, a straightforward argument involving elementary linear algebra shows that the only finite-dimensional seminormed spaces are those arising as the product space of a normed space and a space with trivial seminorm. Consequently, many of the more interesting examples and applications of semi normed spaces occur for infinite-dimensional vector spaces.


See also


  Results from FactBites:
 
PlanetMath: normed vector space (249 words)
It follows that any normed space is a locally convex topological vector space, in the topology induced by the metric defined above.
norm, metric induced by a norm, metric induced by the norm, induced norm
This is version 10 of normed vector space, born on 2002-01-24, modified 2006-12-08.
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