2.5. COMPACTNESS AND CONTINUOUS FUNCTIONS 55

2.5.4 Convergence of FunctionsFirst is the definition of a Banach space. It is really just a generalization of the familiar Rp.For many things of interest in this book, the Banach space will be Rp or Cp.

Definition 2.5.31 A Banach space is a complete normed linear space. It can beeither real or complex, depending on the field of scalars. That is, it is a normed vectorspace in which every Cauchy sequence converges. In particular, it is a metric space inwhich d (x,y)≡ ∥x−y∥ so all the theory of metric space applies. In particular open ballsreally are open.

Here the discussion is specialized to vector valued functions having values in someBanach space X . Most if not all of it will work for general metric spaces.

There are two kinds of convergence for a sequence of functions described in the nextdefinition, pointwise convergence and uniform convergence. Of the two, uniform conver-gence is far better and tends to be the kind of convergence most encountered in complexanalysis. Pointwise convergence is more often encounted in real analysis and necessitatesmuch more difficult theorems.

Definition 2.5.32 Let X ,Y be Banach spaces where ∥·∥ will denote the norm ineither one. S⊆X and let fn : S→Y for n= 1,2, · · · . Then {fn} is said to converge pointwiseto f on S if for all x ∈ S,

fn (x)→ f(x) , that is limn→∞∥fn (x)− f(x)∥= 0

for each x. The sequence is said to converge uniformly to f on S if

limn→∞

(supx∈S∥fn (x)− f(x)∥

)= 0

supx∈S ∥fn (x)− f(x)∥ is denoted as∥fn− f∥∞

or just ∥fn− f∥ for short.∥·∥ is called the uni-form norm. More generally, it suffices in the above to let S just be a metric space.

The following picture illustrates the above definition.

The wriggly function is uniformly close to the not so wriggly one.To illustrate the difference in the two types of convergence, here is a standard example.

Example 2.5.33 Let

f (x)≡{

0 if x ∈ [0,1)1 if x = 1

Also let fn (x) ≡ xn for x ∈ [0,1] . Then fn converges pointwise to f on [0,1] but does notconverge uniformly to f on [0,1].