1-lecture. Normed space. Banach space lesson Plan

EXAMPLES
1. Euclidean space and unitary space . They are Banach spaces with norm defined by
(1.1.3)
In fact, and are complete and (1.1.3) yields the metric (7) in 4-Lecture:

We note in particular that in we have

This confirms our previous remark that the norm generalizes the elementary notion of the length of a vector.
2. Space . It is a Banach space with norm given by

In fact, this norm induces the metric in 4-Lecture (section 4.3):

is a complete space.
3. Space . This space is a Banach space since its metric is obtained from the norm defined by



and it is a complete space.
4. Space . This is a Banach space with norm given by

where .


5. An incomplete normed space and its completion . The vector space of all continuous real-valued functions on forms a normed space with norm defined by
(1.1.4)

This space is not complete. The space can be completed. The completion is denoted by . This is a Banach space. In fact, the norm on and the operations of vector space can be extended to the completion of .

More generally, for any fixed real number , the Banach space

is the completion of the normed space which consists of all continuous real-valued functions on , as before, and the norm defined by

(1.1.5)

The subscript is supposed to remind us that this norm depends on the choice of , which is kept fixed. Note that for this equals (1.1.4).

6. Space s. Can every metric on a vector space be obtained from a norm? The answer is no. A counterexample is the space In fact, is a vector space, but its metric defined by

cannot be obtained from a norm. This may immediately be seen from the following lemma which states two basic properties of a metric obtained from a norm. The first property, as expressed by (1.1.6a), is called the translation invariance of .