# 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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