Convergence in Hilbert's metric and convergence in direction
Let K be a closed pointed cone in a normal linear space X, and for any vectors x,y in K-\(\{\) 0\(\}\), define \(M(x,y)=\inf \{\lambda \geq 0:\lambda y-x\in K\}, m(x,y)=\sup \{\lambda \geq 0:x-y\in K\},\) and \(d(x,y)=\log [M(x,y)/m(x,y)].\) The following result is presented. Theorem. Let (K,d) be as above. Then ''\(d(x_ n,y_ n)\to 0\) implies \(\| x_ n\!'-y_ n\!'\| \to 0\)'' iff K is normal. If the latter condition holds and X is complete then so is \(K'=\{x\in K:\| x\| =1\}.\) Here \(x'=x/\| x\|,\) \(d(\cdot,\cdot)\) is called the Hilbert distance, and K is normal means, \(\inf \{\| x+y\|:x,\quad y\in K'\}>0.\)
scientific article; zbMATH DE number 3862936