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.
Throughout this chapter, we suppose that \(a, b \in \R\) with \(a \lt b\text{.}\) Then the interval \([a,b]\) is a metric subspace of \(\R\text{.}\) We have already seen the space \(C^0([a,b])\) in Example 1.13. In this section, we denote its norm by
