plz give a complete answer for ALL PARTS In these exercises

plz give a complete answer for ALL PARTS

In these exercises, C^0 = C^0([a, b], R) is the space of continuous real-valued functions defined 011 the closed interval [a, b]. It is equipped with the sup norm, ||f|| = sup{f/(x)|: X [a, 6]}. Question Let M, N be metric spaces. (a) Formulate the concepts of pointwise convergence and uniform convergence for sequences of functions f_n: M rightarrow N. (b) For which metric spaces are the concepts equivalent?

Solution

1. Pointwise Convergence of a Sequence

Let E be a set and Y be a metric space. Consider functions fn : E Y for n = 1, 2, . . . . We say that the sequence (fn) converges pointwise on E if there is a function f : E Y such that fn(p) f(p) for every p E. Clearly, such a function f is unique and it is called the pointwise limit of (fn) on E. We then write fn f on E. For simplicity, we shall assume Y = R with the usual metric.

2. Uniform Convergence of a Sequence

Let E be a set and consider functions fn : E R for n = 1, 2, . . . . We say that the sequence (fn) of functions converges uniformly on E if there is a function f : E R such that for every > 0, there is n0 N satisfying

n n0, p E = |fn(p) f(p)| < .

Note that the natural number n0 mentioned in the above definition may depend upon the given sequence (fn) of functions and on the given positive number , but it is independent of p E. Clearly, such a function f is unique and it is called the uniform limit of (fn) on E. We then write fn f on E. Obviously, fn f on E = fn f on E, but the converse is not true : Let E := (0, 1] and define fn(x) := 1/(nx + 1) for 0 < x 1. If f(x) := 0 for x (0, 1], then fn f on (0, 1], but fn 6 f on (0, 1]. To see this, let := 1/2, note that there is no n0 N satisfying

|fn(x) f(x)| = 1 nx + 1 < 1 2 for all n n0 and for all x (0, 1],

since 1/(nx + 1) = 1/2 when x = 1/n, n N.

A sequence (fn) of real-valued functions defined on a set E is said to be uniformly Cauchy on E if for every > 0, there is n0 N satisfying

m, n n0, p E = |fm(p) fn(p)| < .

b)METRIC SPACES

Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one 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 the one 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 suprisingly similar to those we have already seen in one 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 cover 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 number, 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.