Let (X, dx) and (Y, dy) be metric spaces. An isometry is a function f :...

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Let (X, dx) and (Y, dy) be metric spaces. An isometry is a function f : X →Y that preserves distances, i.e., for all x,x' in X, dy(f(x), f(x')) = dx(x,x'). (1) We'll show below that every isometry is an injection. The metric spaces (X,dx) and (Y, dy) are called isometric iff there exists an isometry from X onto Y. 3. Let (X,dx) be a metric space and assume that there exists a positive con- stant C such that, for all x,x in X, dx(x,x)<C. (Equivalently, X has finite diameter.) For each x in X we define the function fx :X → R by f:(t) = d(x,t). Prove that each fr is a bounded and continuous function. (R is equipped with the standard metric. The meaning of 'bounded' is ex- actly what you would expect it to be: a function h : X → R is said to be bounded iff there exists a positive M such that, for all t in X, |h(t)|< M.) 4. Let (X, dx) be as in Problem 3 and let Y be the set of all bounded continuous functions from X to R. Define the metric dy : Y x Y →R by dy (f,8)= sup{|f(t) – g(t)| :1 € X}. (2) For each x in X, let få be as in Problem 3. Prove that the function from X to Y that sends each x to fa is an isometry, i.e., for all x,x in X, dy (fx, fx) = dx(x,x'). (3) Remark: As a consequence of the result in Problem 4, X is isometric to a subspace of Y (the range of the function x+ f). Intuitively, Y contains an exact copy of X. This exact copy will be useful to us when we construct the completion of X, a metric space in which every Cauchy sequence has a limit and contains X as a dense subset. Let (X, dx) and (Y, dy) be metric spaces. An isometry is a function f : X →Y that preserves distances, i.e., for all x,x' in X, dy(f(x), f(x')) = dx(x,x'). (1) We'll show below that every isometry is an injection. The metric spaces (X,dx) and (Y, dy) are called isometric iff there exists an isometry from X onto Y. 3. Let (X,dx) be a metric space and assume that there exists a positive con- stant C such that, for all x,x in X, dx(x,x)<C. (Equivalently, X has finite diameter.) For each x in X we define the function fx :X → R by f:(t) = d(x,t). Prove that each fr is a bounded and continuous function. (R is equipped with the standard metric. The meaning of 'bounded' is ex- actly what you would expect it to be: a function h : X → R is said to be bounded iff there exists a positive M such that, for all t in X, |h(t)|< M.) 4. Let (X, dx) be as in Problem 3 and let Y be the set of all bounded continuous functions from X to R. Define the metric dy : Y x Y →R by dy (f,8)= sup{|f(t) – g(t)| :1 € X}. (2) For each x in X, let få be as in Problem 3. Prove that the function from X to Y that sends each x to fa is an isometry, i.e., for all x,x in X, dy (fx, fx) = dx(x,x'). (3) Remark: As a consequence of the result in Problem 4, X is isometric to a subspace of Y (the range of the function x+ f). Intuitively, Y contains an exact copy of X. This exact copy will be useful to us when we construct the completion of X, a metric space in which every Cauchy sequence has a limit and contains X as a dense subset.

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