Given three vectors $v\_1,v\_2$ and $v\_3$ in a Hilbert space Hthe follwing is true
$$\angle(v\_1,v\_2)+\angle(v\_2,v\_3)\geq \angle(v\_1,v\_3).$$


It tried to substitute $\angle(v\_1,v\_2) = cos^{-1}\frac{v\_1 \cdot v\_2}{\Vert v\_1 \Vert \Vert v\_2 \Vert}$ but I could not show the resulting inequality. What is the name of the inequality and do you know reference that one can cite in an article?



Sum of Angles in a Hilbert space
WLOG we can assume that we are in three space as mentioned above and that the vectors have length $ 1$. With the origin $O$ they span a triangular prism $OABC$. Then the area of say $OAB$ is less than the sum of that of $OAC$ and that of $OCB$. If we express these areas in term of the sines of the angles, we get an inequality which soon gives the required result.


