1-lecture. Normed space. Banach space lesson Plan

1-LECTURE. NORMED SPACE. BANACH SPACE
Lesson Plan:

1. 
   The Сoncept of Normed and Banach Spaces.
   
2. 
   Further Properties of Normed Spaces.
   
3. 
   Finite Dimensional Normed Spaces and Subspaces.
   

1.1. THE СONCEPT OF NORMED AND BANACH SPACES

The examples in the last section illustrate that in many cases a vector space may at the same time be a metric space because a metric is defined on . However, if there is no relation between the algebraic structure and the metric, we cannot expect a useful and applicable theory that combines algebraic and metric concepts.

To guarantee such a relation between "algebraic" and "geometric" properties of we define on a metric in a special way as follows. We first introduce an auxiliary concept, the norm (definition below), which uses the algebraic operations of vector space. Then we employ the norm to obtain a metric that is of the desired kind. This idea leads to the concept of a normed space. It turns out that normed spaces are special enough to provide a basis for a rich and interesting theory, but general enough to include many concrete models of practical importance. In fact, a large number of metric spaces in analysis can be regarded as normed spaces, so that a normed space is probably the most important kind of space in functional analysis, at least from the viewpoint of present-day applications. Here are the definitions:

Definition 1.1.1 (Normed space, Banach space). A normed space is a vector space with a norm defined on it. A Banach space is a also called a normed vector space or normed linear space.

Here a norm on a (real or complex) vector space is a real-valued function on whose value at an is denoted by

and which has the properties

here and are arbitrary vectors in and is any scalar.

A norm on defines a metric on which is given by

(1.1.1)

and is called the metric induced by the norm. The normed space just defined is denoted by or simply by .

The defining properties (N1) to (N4) of a norm are suggested and motivated by the length of a vector in elementary vector algebra, so that in this case we can write . In fact, (N1) and (N2) state that all vectors have positive lengths except the zero vector which has length zero. (N3) means that when a vector is multiplied by a scalar, its length is multiplied by the absolute value of the scalar. (N4) is illustrated in Fig. 1.1.1. It means that the length of one side of a triangle cannot exceed the sum of the lengths of the two other sides.

It is not difficult to conclude from (N1) to (N4) that (1.1.1) does define a metric. Hence normed spaces and Banach spaces are metric spaces.


Fig. 1.1.1. Illustration of the triangle inequality (N4)

Banach spaces are important because they enjoy certain properties which are not shared by incomplete normed spaces.

For later use we notice that (N4) implies

(1.1.2)

The norm is continuous, that is, is a continuous mapping of into .