How Many Are Metric Spaces?

Calculus Level 3

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

$M = \mathbb{R}^n$ and $d\big((x_1, \ldots, x_n), (y_1, \ldots, y_n)\big) = \max_{1\le i \le n} |x_i - y_i|$
$M = \{a, b, c, d\},$ 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 $d(f,g) = \int_{0}^{1} \big(f(x) - g(x)\big)^2 \, dx$

2 1 4 3

1 solution

Daniel Kazmarek
Oct 17, 2020

Maybe I can give you intuition.

For first case, just think of each coordinate in each braces as heights placed side by side as in bar graphs . Hence you get two graphs, superimpose them. Now get the max difference. Think of symmetry preservation and then as for triangle inequality, introduce an other one(think of the upper part of heights), now clearly, the maxes wrt two preexisting graph will be greater in the sum than the original one.

Next, Simply construct a graph with edges and vertices (a,b,c,d) and check it yourself by trial

Third, it is pretty same as the first one, the x belonging to a certain range doe not matter I guess. Lastly, Draw curve and shade areas between them and do the same trick if introducing a third party for detour s I did with the first case. Well, but the answer will be no in this case. See if you can find it yourself.

To be honest, metric spaces are way complicated, so just don't dammn yourself. Correct me if I have any error. I will be glad!

How Many Are Metric Spaces?

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