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In reading I came across the claim that the following is a metric.

For the space $X$ of all integrable functions on the interval [$0,1$] , for $f, g \in X$, the following equation defines a metric:

$\rho(f, g) = \int_0^1 |(f(x) - g(x))| dx$

It is clear to me that $\rho(f, g) = \rho(g, f)$ and also $\rho(f, g) = 0$ if and only $ f = g$. However, does the triangle inequality hold?

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    Triangle inequality follow from triangle inequality for real numbers. In which sense do you mean integrable? Riemann? Lebesgue? (in the last case we can have $\rho(f,g)=0$ even if $f=g$).2012-04-19

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