EDB — 0VT

Exercises

1. [0VT] Prerequisites:[0VS]↺↻.Let \(X=C^ 0([0,1])\) be the space of continuous and bounded functions \(f:[0,1]→ℝ\), endowed with the usual distance

   \[ d_∞(f,g)=\| f-g\| _∞=\sup _{x∈[0,1]}|f(x)-g(x)|\quad . \]

   We know that \((X,d_∞)\) is a complete metric space. Let

   \[ D(0,1)=\{ f∈ X: d_∞(0,f)≤ 1 \} = \{ f∈ X: ∀ x∈[0,1],\quad |f(x)|≤ 1 \} \]

   the disk of center \(0\) (the function identically zero) and radius 1. We know from [0PY]↺↻ that it is closed, and therefore it is complete. Show that \(D\) is not totally bounded by finding a sequence \((f_ n)⊆ D\) as explained in [0VS]↺↻.