EDB — 0NM

Exercises

1. [0NM]Prerequisites:[0B9]↺↻, [192]↺↻,[0N6]↺↻.Difficulty:*.Note:Exercise 2, written exam, 9 July 2011.

   Let \(𝛼(x)\) be a continuous function on \(ℝ\), bounded and strictly positive. Given \(f,g\) continuous functions on \(ℝ\), we define

   \[ d(f,g)=\sup _{x∈ℝ}\big(\min \{ 𝛼(x),|f(x)-g(x)|\} \big)\ . \]

   Prove that \(d\) is a distance on \(C(ℝ)\) and that \(\big(C(ℝ),d\big)\) is complete.

   Solution 1

   [0NP]↺↻