EDB — 0WG

Exercises

1. [0WG]Prerequisites:[0WD]↺↻.We want to define a distance for the space of sequences. We proceed as in [0W9]↺↻. We choose \(X_ i=ℝ\) for each \(i\) and set that \(d_ i\) is the Euclidean distance; then for \(f,g:ℕ→ℝ\) we define

   \[ d(f,g) =∑_ k 2^{-k}𝜑(|f(k)-g(k)|) ~ . \]

   We have constructed a metric space of sequences \((ℝ^ℕ,d)\).

   In the space of sequences \((ℝ^ℕ,d)\) we define

   \[ K=\{ f∈ℝ^ℕ, ∀ k, |f(k)|≤ 1 \} \quad . \]

   Show that \(K\) is compact.

   Solution 1

   [0WH]↺↻