Examples of linear vector spaces

In this. section we analyze some examples of the linear vector spaces, presented in Table Al. 

The simplest example of a linear space is Euclidean space E, which we analyzed above. 

The next example can be constructed using a set of real functions, continuously differentiable to order n on the real interval [a, 6]. Obviously, the sum of two 

There are many different ways to introduce a norm in a function space. The simplest one is as follows  

It is easy to verify that the norm introduced in (A.59) satisfies all required conditions for the norm of the normed space, and I leave this proof to the reader as an exercise. This norm is called the uniform norm. The linear space of continuous functions, Co [a, 6], equipped with the uniform norm, forms a normed space, denoted by C [a, b]. The distance between two functions, /(x) and g x), in the space C[a,b] with the uniform norm is equal to 

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