3. Some function spaces.

Example 1: C[a,b] is the set of continuous functions y=y(x) on [a,b]. To be able to compare different functions we need to define a norm on the space. We can for instance use

Example 2: C¹[a,b] is the set of continuous functions on [a,b] that also has continuous derivative. As norms we can here use the weak norm

Example 3: The spaces Cⁿ[a,b], n=2,3,… ar defined in a corresponding way.

<>Given a specific norm, we get the distance d(y₁,y₂) between two functions y₁ and y₂ as

Remark: The examples 1-3 are examples of normed linear spaces (see also part 11).