How Many Are Metric Spaces?

Calculus Level 5

How many of the following pairs \((M, d)\) are metric spaces?

  • \(M = \mathbb{R}^n\) and \[d((x_1, \ldots, x_n), (y_1, \ldots, y_n)) = \max_{1\le i \le n} |x_i - y_i|\]

  • \(M = \{a, b, c, d\}\) where \(d(a,b) = d(a,c) = 3\), \(d(a,d) = d(b,c) = 7\), and \(d(b,d) = d(c,d) = 11\).

  • \(M = \mathcal{C}[0,1]\), the set of continuous functions \([0,1] \to \mathbb{R}\), and \[d(f,g) = \max_{x\in [0,1]} |f(x) - g(x)|\]

  • \(M = \mathcal{C}[0,1]\) and \[d(f,g) = \int_{0}^{1} (f(x) - g(x))^2 \, dx\]

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