Chapter 9

Basic Function SpacesIn this chapter is an introduction to some of the most important vector spaces of functions.First of all, recall from linear algebra that if you have any nonempty set S and V is the setof all functions defined on S having values in F or more generally some vector space, thendefining

( f +g)(x)≡ f (x)+g(x)

(αg)(x)≡ αg(x)

this defines vector addition and scalar multiplication of functions. You should check thatall the axioms of a vector space hold for this situation. Note also that the usual situation inlinear algebra Fn where vectors are ordered lists of numbers is a special case. There youare considering functions mapping {1, · · · ,n} to F so the set S consists of the first n naturalnumbers. This was a finite dimensional vector space, but if S is the unit interval and Vconsists of functions defined on S, then this will not be finite dimensional because for eachx ∈ S, you could consider fx (x)≡ 1 and fx (y) = 0 for y ̸= x and you would have infinitelymany vectors such that every finite subset of them is linearly independent.

There are two kinds of function spaces discussed here, the space of bounded contin-uous functions and the Lp spaces. First I will consider the space of bounded continuousfunctions.

9.1 Bounded Continuous FunctionsAs before, F will denote either R or C.

Definition 9.1.1 Let T be a subset of some Fm, possibly all of Fm. Let BC (T ;Fn)denote the bounded continuous functions defined on T .1 Then this is a vector space (lin-ear space) with respect to the usual operations of addition and scalar multiplication offunctions. Also, define a norm as follows:

∥f∥ ≡ supt∈T|f(t)|< ∞.

This is a norm because it satisfies the axioms of a norm which are as follows:

∥f+g∥ ≤ ∥f∥+∥g∥ , ∥αf∥= |α|∥f∥

∥f∥ ≥ 0 and equals 0 if and only if f = 0

A sequence {fn} in BC (T ;Fn) is a Cauchy sequence if for every ε > 0 there exists Mε suchthat if m,n≥Mε , then

∥fn− fm∥< ε

Such a normed linear space is called complete if every Cauchy sequence converges. Sucha complete normed linear space is called a Banach space. This norm is often denoted as∥·∥

∞.

I am letting T be a subset of Fn just to keep things in familiar territory. T can be anarbitrary metric space or even a general topological space.

Now consider the general case where T is just some set.1In fact, they will be automatically bounded if the set T is a closed interval like [0,T], but the considerations

presented here will work even when a compact set is not being considered.

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