Normed vector space Totally Explained

Everything about Normed Linear Space totally explained

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.

The zero vector, 0, has zero length; every other vector has a positive length.
Multiplying a vector by a positive number changes its length without changing its direction. See unit vector.
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.

Definition

A semi normed vector space is a pair (V,p) where V is a vector space and p a semi norm on V.
A normed vector space is a pair (V,||·||) where V is a vector space and ||·|| a norm on V.
We often omit p or ||·|| and just write V for a space if it's clear from the context what (semi) norm we're 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:

The vector addition + : V × V → V is jointly continuous with respect to this topology. This follows directly from the triangle inequality.

The scalar multiplication · : K × V → V, 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.
   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 (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 locally compact if and only if the unit ball B =

For each p this defines the same topological space.
   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 seminormed spaces occur for infinite-dimensional vector spaces.

Further Information

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