Some Function Spaces

Chapter 3 Some Function Spaces

One of the reasons why we study metric, normed, and inner product spaces in such an abstract way is that this allows us to treat complicated objects, such as a function, the same way as a point in a Euclidean space, say. Therefore, we can apply similar principles and analyse functions and other objects with familiar tools. The purpose of this chapter is to provide examples of spaces with a view to such applications.

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Throughout this chapter, we suppose that

a, b \in R

with

a < b .

Then the interval

[a, b]

is a metric subspace of

R .

We have already seen the space

C^{0} ([a, b])

in Example 1.13. In this section, we denote its norm by

‖ f ‖_{C^{0} ([a, b])} = sup_{x \in [a, b]} | f (x) |, f \in C^{0} ([a, b]),

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because this makes the notation consistent with other norms that we will introduce shortly.

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