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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. It turns out that the following properties of "vector length" are the crucial ones.

a vector always has a strictly positive length. The only exception is the zero vector which has length zero.
multiplying a vector by a positive number has the same effect on the length.
the triangle inequality, which amounts roughly to saying that the distance from A through B to C is never shorter than going directly from A to C. I.e. 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.

Contents

1 Definition

2 Topological structure

3 Linear maps and dual spaces

4 Normed spaces as quotient spaces of semi normed spaces

5 Finite product spaces

6 See also

Definition

A semi normed vector space is a 2-tuple (V,p) where V is a vector space and p a semi norm on V.

A normed vector space is a 2-tuple (V,||&middot||) where V is a vector space and ||&middot|| a norm on V.

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

Topological structure

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 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 by the semi-norm.

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.

All norms on a finite-dimensional vector space are equivalent from a topological point as they induce the same topology. 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.

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

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

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.

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.

The norm is a continuous linear transformation and all linear maps between finite dimensional vector spaces 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.

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.

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 L^p spaces, the function defined by

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

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.

Finite product spaces

Given n semi normed spaces X_i with semi norms p_i we can define the product space as

$X := \prod_{i=1}^{n} X_i$

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 mulitplication defined as

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

We define a new function p

$p:X \mapsto \mathbb{R}$

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

which is a semi norm on X. The function p is a norm if and only if all p_i 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.

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