How Many Are Metric Spaces?

Calculus Level 3

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

  • M=RnM = \mathbb{R}^n and d((x1,,xn),(y1,,yn))=max1inxiyid\big((x_1, \ldots, x_n), (y_1, \ldots, y_n)\big) = \max_{1\le i \le n} |x_i - y_i|

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

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

  • M=C[0,1]M = \mathcal{C}[0,1] and d(f,g)=01(f(x)g(x))2dxd(f,g) = \int_{0}^{1} \big(f(x) - g(x)\big)^2 \, dx

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