**Question:** Two $k$-dimensional subspaces $W\_1,W\_2$ with associated orthogonal projections $P\_1, P\_2$ are isoclinic with parameter $\lambda \ge 0$ if $P\_1P\_2P\_1=\lambda P\_1$ and $P\_2P\_1P\_2=\lambda P\_2$. 


I was able to show that if $W\_1\perp W\_2$, then $W\_1,W\_2$ are isoclinic with $\lambda=0$. Now I've read the following definitions in a paper. 


* Two subspaces $W\_1,W\_2$ are $\epsilon$-nearly orthogonal if for all unit vectors $\phi\in W\_1$ and $\psi\in W\_2$ we have $|\langle\phi,\psi\rangle|<\epsilon$.
* Two k-dimensional subspaces $W\_1,W\_2$ are $\epsilon$-nearly isoclinic if there exists $\lambda\geq0$ such that $(\lambda-\epsilon^2)P\_1\le P\_1P\_2P\_1\le(\lambda+\epsilon^2)P\_1$ and
$(\lambda-\epsilon^2)P\_2\le P\_2P\_1P\_2\le(\lambda+\epsilon^2)P\_2$.


Then the claim is again, if $W\_1,W\_2$ have same dimension, then $\epsilon$-nearly orthogonal implies $\epsilon$-nearly isoclinic with parameter $\lambda=0$. In the paper they treat it like it is trivial. But for me, regrettably, it is not. Is there really a short proof for that or does anybody know where i can find a proof for that (if it is not so short)?



$\epsilon$-nearly isoclinic
Hey I just found a solution on my own.  

Let $a$ be an arbitrary Element of a Hilbert space.  

We have  

$\Vert P\_2 P\_1 a\Vert^2=\langle P\_2 P\_1a,P\_1a\rangle=\Vert P\_2 P\_1 a\Vert\Vert P\_1a\Vert\langle \frac{P\_2P\_1a}{\Vert P\_2 P\_1 a\Vert},\frac{P\_1a}{\Vert P\_1 a\Vert}\rangle<\epsilon\Vert P\_2 P\_1 a\Vert\Vert P\_1a\Vert$.  

$\implies \Vert P\_2 P\_1 a\Vert<\epsilon\Vert P\_1a\Vert$  

$\implies\Vert P\_2 P\_1 a\Vert^2<\epsilon^2\Vert P\_1a\Vert^2$  

$\implies\langle P\_1P\_2P\_1a,a\rangle\le\epsilon^2\langle P\_1a,a\rangle$.  

Since  

$-\epsilon\Vert P\_1a\Vert^2<-\Vert P\_2 P\_1a\Vert^2<\Vert P\_2 P\_1a\Vert^2$,  

we also have  

$-\epsilon^2\langle P\_1a,a\rangle\le\langle P\_1P\_2P\_1a,a\rangle$.  

Changing the roles of $P\_1$ and $P\_2$ yields the claim. 


