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.


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