I was trying to demonstrate that the space l^p is a metric space.
Let $p \geq 1$ a fixed real number. By definition, each element in the space $l^p$ is a sequence $x=(\xi_j)=(\xi_1,\xi_2,...)$ of numbers such that
$$\sum_{j=1}^{\infty}|\xi_j|^p<\infty$$
define the metric by $$d(x,y)=\left( \sum_{i=1}^{\infty} |\xi_i-\delta_i|^p \right)^{\frac{1}{p}}$$
where $x=(\xi_j)$ and $y=(\delta_j)$
Of course all axioms are trivial except the triangular inequality.
If I suppose that
$$\left( \sum_{i=1}^{\infty} |\xi_i+\delta_i|^p \right)^{\frac{1}{p}} \leq \left( \sum_{i=1}^{\infty} |\xi_i|^p \right)^{\frac{1}{p}}+\left( \sum_{i=1}^{\infty} |\delta_i|^p \right)^{\frac{1}{p}}$$
holds, this implies that the triangular inequality holds and the metric converges. So my goal is to prove that inequality. This is Minkowski inequality.
I'm not sure about that but, apparently, Minkowski proved this inequality in 1901. I do not know how he do it, I suppose that He used the Holder inequality, apparently Holder proved that in 1889, so its possibly that Minkowski was aware of the Hölder inequality. This makes the proof a lot of simple. What is the context? Why did Hölder arrive at this inequality? I want that information to construct that proof intuitively. Can someone tells me why Holder came up with that inequality? Thats the Holder inequality (I know how to proof that, but I don't have the intuition)
$$\sum_{i=1}^{\infty}|\xi_i \delta_i| \leq \left( \sum_{i=1}^{\infty} |\xi_i|^p \right)^{\frac{1}{p}}\left( \sum_{i=1}^{\infty} |\delta_i|^p \right)^{\frac{1}{p}}$$
