# Triangle Inequality for Angles in Projective Space

I want to show that the angle between two lines through the origin in a (complex or real) inner product vector space $$(V,\langle \cdot,\cdot\rangle)$$ is a distance function which turns $$\mathbb{P}V$$, the projective space of $$V$$ (the lines through the origin), into a metric space. I found this post (metric and measure on the projective space) in which they say this is in fact a distance, but no reference is given. I also found a this link (Triangle inequality for angles), which is for $$\mathbb{R}^3$$, but the answer is kind of confusing and it is not appropriate for the complex case, I believe.

I understand the angle between two lines $$l,l'$$ through the origin as the the number $$\sphericalangle(l,l') := \cos^{-1} \left(\frac{|\langle v,v'\rangle|}{|v||v'|} \right),$$ in which $$v \in l$$, $$v' \in l'$$. This does not depend on the choice of $$v,v'$$, and they can in fact be chosen such that $$|v|=|v'|=1$$.

The first two properties of distances are pretty easy, so I just need to show the triangle inequality. For three lines $$l,l',l''$$, we must show that $$\sphericalangle(l,l'') \leq \sphericalangle(l,l') + \sphericalangle(l',l''),$$ For $$v \in l$$, $$v' \in l'$$, $$v'' \in l''$$, that is $$\cos^{-1} \left(\frac{|\langle v,v''\rangle|}{|v||v''|} \right) \leq \cos^{-1} \left(\frac{|\langle v,v'\rangle|}{|v||v'|} \right) + \cos^{-1} \left(\frac{|\langle v',v''\rangle|}{|v'||v''|} \right)$$

What I have so far is similar to what is explained in the second link above. I projected $$v'$$ in the subspace spanned by $$v$$ and $$v''$$, to get the vector $$w$$. Therefore I must prove that

1. $$\cos^{-1} \left(\frac{|\langle v,w\rangle|}{|v||w|} \right) \leq \cos^{-1} \left(\frac{|\langle v,v'\rangle|}{|v||v'|} \right)$$. For this one, I wish to use the fact that $$\cos^{-1}\colon [0,1] \to [0,\pi/2]$$ is decreasing, so I must prove that $$\frac{|\langle v,w\rangle|}{|v||w|} \geq \frac{|\langle v,v'\rangle|}{|v||v'|};$$
2. The triangle inequality holds for $$v,w,v''$$, which are coplanar.

I would like some tips on how to proceed on both items, and any additional commentary would be awesome. I would also like to know if this is true for infinite dimensional $$V$$, and a proof that does not use a basis would be perfect.