EDB — 0R5

Exercises

1. [0R5] Let \((X,d)\) be a metric space where \(X\) is also a group. Let \(Θ\) be a subgroup.

   We define that \(x∼ y\iff x y^{-1}∈Θ\). It is easy to verify that this is an equivalence relation. Let \(Y=X/∼\) be the quotient space. ¹

   Suppose \(d\) is invariant with respect to left multiplication by elements of \(Θ\):

   \begin{equation} d(x,y)=d(𝜃 x,𝜃 y)~ ~ ∀ x,y∈ X,∀ 𝜃∈Θ~ ~ . \label{eq:d_ invarian_ grupp} \end{equation}

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   (This is equivalent to saying that, for every fixed \(𝜃∈Θ\) the map \(x↦ 𝜃 x\) is an isometry). We define the function \(𝛿:Y^ 2→ℝ\) as in [(9.57)]↺↻.

   • Show that, taken \(s,t∈ X\),

     \begin{equation} 𝛿([s],[t]) = \inf \{ d(s,𝜃 t) : 𝜃 ∈ Θ \} \label{eq:dist_ quoz_ 1gruppo} \end{equation}

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     where \([s]\) is the class of elements equivalent to \(s\).

   • Show that \(𝛿≥ 0\), that \(𝛿\) is symmetric and that \(𝛿\) satisfies the triangle inequality.

   • Suppose that, for every fixed \(t∈ X\), the map \(𝜃↦ 𝜃 t\) is continuous from \(Θ\) to \(X\); suppose also that \(Θ\) is closed: then \(𝛿\) is a distance. ²

   Solution 1

   [0R6]↺↻