## Sum of Angles in a Hilbert space [closed]

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?

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You can assume the Hilbert space is $\mathbb R^3$ (spanned by the vectors). – Anthony Quas Sep 5 at 13:12

## closed as off topic by Anthony Quas, Jon Bannon, Andreas Blass, Bill Johnson, S. SraSep 5 at 17:00

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.
The angle (in $[0,\pi]$) corresponds to geodesic distance on the unit sphere. This is just the triangle inequality. No reference needed.