Metric Spaces  Iteration and Application : Summary
Instead of approaching “Metric spaces” from a real analysis perspective, this book tries to use an application to motivate the reader. The application is based on “contraction mapping theorem” which is used in solving a single equation / simultaneous equations. In its most simple form, let’s say there is an equation of the type x = f(x) which needs to be solved. One of the ways to solve this type of equation is as follows:
Start with an initial value, x1, and then calculate x2= f(x1), x3= f(x2) , x4 = f(x3) , …. etc. The sequence x1, x2, x3,…. converges to the root of the equation.
Now the obvious questions are : Why does this procedure work ? When does the procedure break down ? The root of such an equation is called fixed point as function does not change the value. The principle behind this simple question can be found in the concepts underlying “Metric Spaces”.
Typically one comes across metric spaces in a real analysis text. This book is welcome breather to people who want to know about metric spaces but are rusty on “Real Analysis” fundas. Let me quickly summarize the content of the book.
**Chapter 1 : Sequences by iteration
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The first chapter provides motivation to get a feel of the sequences and their convergence. With the need to solve a simple equation x= f(x), where f(x) is continuous, a few examples are shown where iterative procedure gives the root of the equation. The questions that are asked for each of the examples are :

Will the sequence of x’s that result from iteration ALWAYS converge ? If not, what are the conditions under which the sequence does diverges?

If it does converge , Will the limit definitely be a root of x = f(x)
Iterations need not deal with only numbers. If you take a function x(t) , and apply a transformation (let’s say an integral transformation) , it is likely that the transformation gives back x(t) . Hence there needs to be a framework to solve such problems where a transformation does not the change the function. Basically the first chapter is used to show the reader various examples that use a sequence of numbers / sequence of functions/ to solve fixed point/function problems. One lingering question at the end of this chapter is, “What does it mean to say that a sequence of values / functions converge?”
Chapter 2 : Metric Spaces
The second chapter introduces the concept of metric spaces. The basic definition of a metric d is given below:
Metric space is then symbolized as (X,d). To give a general flavour of metric spaces, the following metric spaces are discussed

X = R , d( x, y ) = xy

X = C, d( x, y ) = xy

X = R^2 , d( x, y ) = Euclidean Distance

X = R^n , d( x1, x2,…, xn ) = Euclidean Distance

Discrete space

X = R^2 and d( x, y ) is lift metric / raspberry pickers’ metric

X is the set of continuous functions from [ a, b ] to R , d( x, y ) is based on max metric

X is the set of bounded functions from [ a, b ] to R , d( x, y ) is based on sup metric

X is the set of continous functions from [ a, b ] to R , d( x, y ) is based on adding up all the distances apart of their graphs.
The examples also show that it is important to think about mathematical objects in terms of their roles than their existential meaning, a point extremely well pointed out by Timothy Gowers in his short intro book on math. Once a metric space is defined with a certain properties, it allows one to talk about metric spaces in a variety of contexts.
In the chapter, subsequently there is a discussion on sequences and Cauchy convergence is discussed. One crucial but often neglected point mentioned at this point of the book is “ Whenever we talk about a sequence , we need to mention the metric space that we are taking about” . Reason being a sequence can converge for a specific metric in R while diverge for a different metric in R.
**Chapter 3 : The three C’s
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The third chapter is titled “The three C’s”, referring to Closed, Complete , Compact properties of a set.This chapter is a little dense but one needs to know these 3 C’s to appreciate Metric spaces as some properties of the sets are invariant under change of metric space. Compactness does not depend on metric spaces. The chapter introduces each of C’s from a sequence angle. The basic idea behind introducing these concepts is to answer the question, “If there are a series of numbers in a sequence, how can one talk about convergence ? ”. A look at Cauchy convergence does not guarantee convergence / uniform convergence. Hence there needs to be additional constraints to take a Cauchy sequence that results out of iterative procedure to call it the root of the equation. What are the three C’s in simple words?
A complete set if one in which if a sequence has the property that the distance between terms tend to zero, then the sequence converges to a point in the set. A complete set is necessarily closed but a closed set need not be complete. Why bother about complete sets? Because these are precisely the sets for which the Cauchy convergence is enough to ensure convergence. If we restrict the attention to complete metric spaces (X,d) then subsets of X are complete if and only if they are closed. The takeaway from this chapter is the following dependency
Compact => Complete => Closed
One of the other takeaways from this chapter is that, if a sequence is bounded and closed, it is compact.
**Chapter 4 : The contraction mapping principle
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This chapter talks about the underlying principles of the contraction mapping theorem. The first chapter is all good as it appeals to the intuition. However one needs to get the math behind it and this chapter explains using the concept of contraction.
Using the above concept, if f be a contraction of complete metric space, then f has a unique fixed point. Another extension to the above definition is when the space is compact metric space. In such a metric space, a weaker condition would suffice, k can be equal to 1.
So, this is where all the concepts come together in the book. Metric spaces, complete sets, compact sets are beautifully tied to the framework for solving x=f(x) equation. The best thing about defining rules about mathematical objects is that it allows you to move in to various dimensions / spaces. The chapter then goes to show the concepts behind the contractual mapping theorem as applied to PDEs and integration involved equations.
Chapter 5 : What makes analysis work ?
The last chapter goes back to real analysis concepts introduced in chapter 2 and tries to reiterate the concepts of open balls, closed balls, uniform continuity, closed sets, compact sets, connected sets. It then applies these concepts to classic theorems in real analysis. So, the last chapter , I think serves as a motivator to the reader to go back to heavy real analysis machinery books like Rudin etc.
Takeaway :
If you want to understand “Metric Spaces” and are rusty with real analysis concepts, this book is perfect. The treatment is different in the sense that it avoids “epsilon/delta” argument and uses sequences to explain the principles behind “Metric Spaces”