I have a two part question: I need to show that $d(f,g)=\int_{-1}^1\! |f(x)-g(x)| \, \mathrm{d}x$ is a metric in $C((-1,1),\mathbb{R)}$ and furthermore prove/disprove that the space $C((-1,1),\mathbb{R)}$ is complete with that very metric.
I don't know how to show that the triangle inequality holds whereas I really can't get anywhere with the Banach space business. I don't even have a guess on whether it is complete with that metric or not so a reference to any specific literature or hints in general are greatly appreciated.
Thank you, ramleren
The triangle inequality is obvious, if you use the fact that the euclidean distance is a metric together with "monotonicity" of the Riemann-integral.
For the completeness i think this function might work as a counterexample $$f_{n}(x) = \left\{ \begin{array}{lr} 0 & : x \in (-1,-\frac{1}{2})\\ (x+\frac{1}{2})^{n} & : x \in [-\frac{1}{2},\frac{1}{2})\\ 1 & : x \in [\frac{1}{2},1) \end{array} \right.$$ I let you sort out the details.