User:Robinh/Sandbox

Hi. I'm reading a maths paper that discusses positive definite functions and am having difficulty interpreting the notation. Here is the quote:

$K(x,y)=f(||x-y||),x\in X$

where $||\cdot ||$ is again the Euclidean norm on ${\mathbb {R} }^{n}$ but $X=S^{n-1}$ is the unit sphere in ${\mathbb {R} }^{n},n\geq 2$. As there is a simple 1-1 correspondence between $||x-y||$ and the standard inner product $<x,y>$ for all $x,y\in S^{n-1}$, it is more convenient to consider $K(x,y)=g(<x,y>)$. Schoenberg proved that, for $g\in C\left[0,1\right]$, blah blah blah.

OK, three questions:

• Is the "simple 1-1 correspondence" just $\arccos(x\cdot y)$?
• What is $C[0,1]$?