NORMED LINEAR SPACES

The vectors in a linear space may have no natural concept of "length" associated with them. For example, the set of all square matrices of order n constitute a linear space yet there is no natural concept of a "length" associated with a matrix. However, we can define "lengths" for all the matrices in the space (i.e. assign a length to each matrix). The "length" we assign is call the "norm". We assign a norm to all vectors in the space. Similarly for the space of functions. Functions have no natural concept of "length" associated with them. But we can assign one (assign a norm).

A linear space for which a "length" (or norm) has been defined is called a normed linear space.

Normed linear space (or normed vector space). An abstract mathematical system that meets the axiomatic requirements for being a linear space becomes a normed linear space if we, by one means or another, assign to each vector of the space a real number ||x||, called the norm, possessing the following properties:

1) ||x|| > 0 if x 0

2) ||ax|| = |a| ||x||

3) ||x + y|| ||x|| + ||y||

Induced metric on a normed linear space. Let V be a normed linear space. The function “d” defined by

            d(u, v) = ||u - v||

where u, v ε V, is a metric (or distance function) on V and is called the induced metric on V.

Thus every normed linear space with the induced metric is a metric space and hence also a topological space. A normed linear space is also a linear topological space.