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Let (X, d) be a metric space, and let U denote the set of all uniformly...

  1. Let (X, d) be a metric space, and let U denote the set of all uniformly continuous functions from X into R. (a) If f,g ∈ U and we define (f + g) : X → R by (f + g)(x) = f(x) + g(x) for all x in X, show that f+g∈U. In words,U is a vector space over R. (b)If f,g∈U and we define (fg) : X → R by (fg)(x) = f(x)g(x) for all x in X, show by an example that it does not necessarily follow that fg ∈ U. If, however, the functions are also bounded, then fg ∈ U. [A function f : X → (Z,ρ) is bounded if f(X) is a bounded subset of Z.] (c) Can you give some conditions under which the quotient of two uniformly continuous functions is uniformly continuous?

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