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[1JX]Prerequisites:[1J3] subpoint [6],[1JN]. Let \(IβŠ† ℝ\) be a subset. Let \(X\) be the set of functions \(f:I→ℝ\) bounded and uniformly continuous. We equip \(X\) with distance \(d_∞(f,g)=\| f-g\| _∞\). Show that the metric space \((X,d_∞)\) is complete.

Solution 1

[1JY]

In particular, \(X\) is a closed vector subspace of the space \(C_ b(I)\) of continuous and bounded functions.

[ [1JZ]]

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Authors: "Mennucci , Andrea C. G." .
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  • convergence, uniform ---
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  • space, of uniformly continuous functions
  • function, uniformly continuous, space of ---
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