10.12 Infinite product of metric spaces

E301

[0W9] Prerequisites:5.Sia \(𝜑(t)=t/(1+t)\). Let \((X_ i,d_ i)\) be metric spaces with \(i∈ℕ\), let \(X=∏_{i∈ℕ} X_ i\), for any \(f,g∈ X\) we define the distance on \(X\) as

\[ d(f,g) =∑_{k=0}^∞ 2^{-k}𝜑(d_ i(f(k),g(k))) ~ . \]

Prove that \(d\) is a distance.

E301

[0WB] Let \(f,f_ n∈ X \) be as before in 1, show that \(f_ n→_ n f\) according to this metric if and only if for every \(k\) we have \(f_ n(k)→_ n f(k)\).

E301

[0WC]Let \((X_ i,d_ i)\) and \((X,d)\) be as before in 1. If all the spaces \((X_ i,d_ i)\) are complete, prove that \((X,d)\) is complete.

E301

[0WD] Prerequisites:5,2.Difficulty:*.Let \((X_ i,d_ i)\) and \((X,d)\) be as before in 1. If all the spaces \((X_ i,d_ i)\) are compact, prove that \((X,d)\) is compact. Hidden solution: [UNACCESSIBLE UUID ’0WF’]

E301

[0WG]Prerequisites:4.We want to define a distance for the space of sequences. We proceed as in 1. 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. Hidden solution: [UNACCESSIBLE UUID ’0WH’]

E301

[0WJ]Let \(N(𝜌)\) be the minimum number of radius balls \(𝜌\) that are needed to cover \(K\) (from the previous exercise 5). Estimate \(N(𝜌)\) for \(𝜌→ 0\).

See also Sec. 11