If $d(x,y)$ and $e(x,y)$ are metrics then $d(x,y)+e(x,y)$ and $\frac{d(x,y)}{1+d(x,y)}$ are metrics.


If $d\_i(x,y)$ for $i=1,\dots,n$ are metrics then so is $\sqrt{\sum\_{i=1}^n{d\_i^2(x,y)}}$


Are there other interesting ways of constructing new metrics from old metrics?



What are some interesting ways of making new metrics out of old metrics?
Personally, I prefer $\min(d(x,y),\epsilon)$ over the standard $d(x,y)/(1+d(x,y))$ trick if the goal is to turn a metric into an equivalent metric – it's a lot easier to prove the metric axioms for this one.


From that follows one way to turn a countably infinite number of metrics into a new metric: $d(x,y)=\sum\\_{n=1}^\infty \min(d\\_n(x,y),2^{-n})$. The point being that convergence in the new metric is the same as convergence in all of the original ones. So this can be used to prove stuff like $C(\mathbb{R})$ with the topology of uniform convergence on compact sets is a metrizable space, and similarly for the space of test functions used in distribution theory. (The individual $d\\_n$ could even be pseudometrics rather than metrics; it doesn't matter, so long as $x\ne y$ implies $d\\_n(x,y)\ne0$ for *some* $n$.)


