Sum of Angles in a Hilbert space - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-23T05:03:27Z http://mathoverflow.net/feeds/question/106430 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106430/sum-of-angles-in-a-hilbert-space Sum of Angles in a Hilbert space warsaga 2012-09-05T13:07:23Z 2012-09-05T16:50:17Z <p>Given three vectors $v_1,v_2$ and $v_3$ in a Hilbert space Hthe follwing is true $$\angle(v_1,v_2)+\angle(v_2,v_3)\geq \angle(v_1,v_3).$$</p> <p>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?</p> http://mathoverflow.net/questions/106430/sum-of-angles-in-a-hilbert-space/106449#106449 Answer by jbc for Sum of Angles in a Hilbert space jbc 2012-09-05T16:45:33Z 2012-09-05T16:45:33Z <p>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.</p> http://mathoverflow.net/questions/106430/sum-of-angles-in-a-hilbert-space/106450#106450 Answer by Robert Israel for Sum of Angles in a Hilbert space Robert Israel 2012-09-05T16:50:17Z 2012-09-05T16:50:17Z <p>The angle (in $[0,\pi]$) corresponds to geodesic distance on the unit sphere. This is just the triangle inequality. No reference needed.</p>