An Analysis upon Various Aspects of Metric Spaces in Real Analysis: Some Solutions

INTRODUCTION

Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in single variable calculus, especially arguments concerning convergence and continuity. The reason is that the notions of convergence and continuity can be formulated in terms of distance, and that the notion of distance between numbers that you need in single variable theory, is very similar to the notion of distance between points or vectors that you need in the theory of functions of severable variables. In more advanced mathematics, we need to find the distance between more complicated objects than numbers and vectors, e.g. between sequences, sets and functions. These new notions of distance leads to new notions of convergence and continuity, and these again lead to new arguments surprisingly similar to those you have already seen in single and several variable calculus. After a while it becomes quite boring to perform almost the same arguments over and over again in new settings, and one begins to wonder if there is general theory that covers all these examples { is it possible to develop a general theory of distance where we can prove the results we need once and for all? The answer is yes, and the theory is called the theory of metric spaces. A metric space is just a set X equipped with a function d of two variables which measures the distance between points: d(x; y) is the distance between two points x and y in X. It turns out that if we put mild and natural conditions on the function d, we can develop a general notion of distance that covers distances between numbers, vectors, sequences, functions, sets and much more. Within this theory we can formulate and prove results about convergence and continuity once and for all. The purpose of this chapter is to develop the basic theory of metric spaces. In later chapters we shall meet some of the applications of the theory. We now introduce the idea of a metric space, and show how this concept allows us to generalise the notion of continuity. We will then concentrate on looking at some examples of metric spaces.

As already mentioned, a metric space is just a set X equipped with a function that measures the distance d(x, y) between points For the theory to work, wc need the function d to have properties similar to the distance functions we are familiar with. So what properties do wc expect from a measure of distance? First of all, the distance d(x, y) should be a nonnegative number, and it should only be equal to zero if x = y. Second, the distance d(x,y) from x to y should equal the distance d(y, x) from y to x. Note that this is not always a reasonable assumption - if we, e.g., measure the distance from x to y by the time it takes to walk from x to y, d(x,y) and d(y,x) may be different - but we shall restrict ourselves to situations where the condition is satisfied. The third condition we shall need, says that the distance obtained by going directly from x to y, should always be less than or equal to the distance we get when we go via a third point z, i.e. It turns out that these conditions are the only ones we need, and we sum them up in a formal definition.

have

Comment: When it is clear - or irrelevant - which metric d we have in mind, we shall often refer to “the metric space X'' rather than “the metric space (X.d)''. Let us take a look at some examples of metric spaces. Example 1: If we let then is a metric space. The first two conditions are obviously satisfied, and the third follows from the ordinary triangle inequality for real numbers: Example 2: If we let thenis a metric space. The first two conditions are obviously satisfied, and the third follows from the triangle inequality for vectors the same way as above : Example 3: Assume that we want to move from one point in the plane to another but that we are only allowed to move horizontally and vertically. If we first move horizontally from to and then vertically from to , the total distance is Also in this case the first two conditions of a metric space are obviously satisfied. To prove the triangle inequality, observe that for any third point we have where we have used the ordinary triangle inequality for real numbers to get from the second to the third line.

Before we look at what it means for a sequence to be convergent with respect to a given metric, we spend a little time discussing one way of gaining some understanding about the geometric meaning of a given metric. In the last subsection, we met three different metrics: the discrete metric, the taxicab metric on the plane and a mixed metric on the plane (which was formed from the usual distance in R together with the discrete metric). An easy way to gain some insight into the behaviour of a metric is to look at the balls around a given point. For the usual Euclidean distance in a ball of radius r around a point consists of all those points whose distance from a is at most r, and this definition naturally extends to general metric spaces. However, in the following definition we take care to distinguish between balls that include points at exactly distance r from the centre a and those that do not.

Let (X, d) be a metric space, and let and The open ball of radius r with centre a is the set The closed ball of radius r with centre a is the set

When r = 1, these sets are called respectively the unit open ball with centre a, the unit closed ball with centre a and the unit sphere with centre a.

Let {X.d) be a metric space, and let . Show that and

It follows from (Ml) that and We now discover what open balls, closed balls and spheres look like for some of the metric spaces we have met already. Let us start by determining the open and closed balls for the discrete metric,

Let X be a non-empty set and . Determine Solution Let and suppose that Since and unless a=x (when it is 0), we conclude that

Consider the metric space - that is, the plane with the taxicab metric. Find the unit open ball The centre is and we want to find all points that satisfy We first consider points in the first quadrant, where We want to find those points where Consider the line or equivalently In the first quadrant, this line connects the points (0,1) and (1,0). The points on this line segment have coordinates All points below the line segment have coordinates with and all points on or above it have coordinates with Hence the points where are those strictly below the line segment, making up the shaded region. By use of a similar argument for each of the other three quadrants, or by appealing to the symmetry of the situation, we obtain triangular regions in each quadrant. Combining these, we obtain the diamond-shaped region; the open ball is the set of points strictly inside this diamond, shown shaded in the figure. The dashed boundary indicates that it is not included in the set.

Now that we have several examples of metric spaces available to us, we return to the problem of defining continuous functions between metric spaces. Since the definition of a general metric space is modelled on the properties of the Euclidean metric on and we defined continuity of functions between Euclidean spaces in terms of convergent sequences, it is natural to attempt to extend our ideas about convergent sequences in to general metric spaces. In fact, we did much of the hard work when we generalised from the notion of convergence for real-valued sequences to that of convergence of sequences init is now only a short step to develop these concepts for the metric space setting. We observed that a real sequence can be thought of as a function given by Note that the only role played by here is as the codomain of the function the structure of becomes relevant only when convergence is considered. Since the codomain of a function is simply a set, the following definition is a natural generalisation.

ordered list of elements of X: The element a* is the kth term of the sequence, and the whole sequence is denoted by

The definition of what it means for a sequence to converge in a metric space (X, d) is closely based on the definition of convergence in

Let (X.d) be a metric space. A sequence (ak) in X d-converges to if the sequence of real numbers is a null sequence. We write as or simply if the context is clear. We say that the sequence is convergent in (X,d) with limit a. A sequence that does not converge (with respect to the metric d) to any point in X is said to be d-divergent..

Let be the plane with the taxicab metric, and let be the sequence given by Show that converges to (1,2) with respect to e1. Convergent sequences in have unique limits - that is, a sequence cannot simultaneously converge to two different limits. The next result establishes this as a fact in any metric space.

Proof We use proof by contradiction. Suppose that the sequence d-converges to both a and b in X, with Thenby property (Ml) of Definition 1.1 for d, and so if we let then Since we are supposing that the sequence converges to both a and b, the sequences of real whenever The Triangle Inequality (property (M3) for d) tells us that, for each , by the definition ofBut this is impossible; hence our initial assumption that a sequence could converge to two distinct limits must be wrong. We conclude that any d-convergent sequence has a unique limit.

Now that we know what it means for a sequence to converge in a metric space, we can formulate a definition of continuity for functions between metric spaces.

Let (X, d) and (Y, e) be metric spaces and let be a function. Then f is (d, e)-continuous at if: Whenever is a sequence in X for which as then the sequence as If f does not satisfy this condition at some - that is, there is a sequence in X for which as but does not converge to then we say that f is (d, e)-discontinuous at a. A function that is continuous at all points of X is said to be (d, e)-continuous on X (or simply continuous, if no ambiguity is possible). Our next worked exercise shows that this definition can make some surprising functions continuous.

Let be a function and let Prove that f is always -continuous at a.

Let and suppose that is a sequence in that is -convergent to a. Then we deduce that there is so that for , But then for and

real null sequence and we conclude that f is -continuous at a. This is a rather artificial example and it tells us that every function from to is -continuous on However, it does illustrate that our intuitive notion of what continuity means breaks down when looking at metrics different from the Euclidean ones, and so highlights the importance of working from the definition.

Let be given by Prove that f is -continuous on

Let We must show that if (x*) is a sequence in the plane that -converges to a, then is a real null sequence. Suppose that is a sequence in the plane that -converges to a, that is, a sequence for which is a real null sequence. Then by definition of f and But for every k and we are assuming that is a real null sequence. Hence by the Squeeze Rule, as That is, is also a real null sequence and so f is -continuous at a. Since was an arbitrary point in we conclude that f is -continuous on At the moment our stock of metric spaces is quite small: Euclidean spaces, the plane with the taxicab metric, the plane with a particular ‘mixed5 metric, and arbitrary sets with the discrete metric. In the next chapter we will look at more examples of metric spaces and examine further the notion of continuity. What we can do at this point, though, is prove a useful result that applies to all continuous functions and which is an extension of the Composition Rule for continuous functions between Euclidean spaces. Cambridge University Press, Cambridge. 2. Gerald B. Folland (1999). Real Analysis: Modern Techniques and Applications, 2nd Edition, John Wiley & Sons. 3. Jeremy Gray (2015). The Real and the Complex: A History of Analysis in the 19th Century, Springer-Verlag. 4. Kenneth R. Davidson, Allan P. Donsig (2009). Real Analysis and Applications, Springer-Verlag. 5. M.K Singal and Asha Rani Singal (2005). Topics in Analysis II (Metric Spaces), R. Chand and Co., New Delhi. 6. Micheal O. Searcoid (2008). Metric Spaces, Springer International Edition, New Delhi. Satish Shirali and Harkrishan L. Vasudeva, Metric Spaces, Springer, 2006. 7. Rodney Coleman (2012). Calculus on Normed Vector Spaces, Springer Verlag.

Assistant Professor in Mathematics, C. R. S. U., Jind, Haryana