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I want to show that if $L:\mathbb{R}^d\rightarrow\mathbb{R}^d$ is a linear isomorphism and $V_1,V_2$ are linear subspaces of $\mathbb{R}^d$ then:

$$\frac{1}{||L||\space||L^{-1}||}\leq\frac{|sin\measuredangle(L(V_1),L(V_2))|}{|sin\measuredangle(V_1,V_2)|}\leq||L||\space||L^{-1}||$$ where the angle $\measuredangle (V,W)$ between two subspaces $V$ and $W$ of $\mathbb {R}^d $ is the smallest angle between non-zero vectors $v\in V$ and $w\in W$.

I tried to solve the first inequality using the relation between the sinus and the vector product but I think that it is not the correct way.

Can someone help me? Thank you.

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    How do you define an angle between subspaces? – 2017-01-03
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    The angle $\measuredangle (V,W)$ between two subspaces $V$ and $W$ of $\mathbb {R}^d $ is the smallest angle between non-zero vectors $v\in V$ and $w\in W$. – 2017-01-03
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    Do you have any idea? – 2017-01-03
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    I'd try using that the norm of the linear map can also be calculated as $$\sup_{v,w} \frac{\langle Lv, Lw \rangle}{\|v\| \|w\|}$$, mix that with the inner product formula for $\cos$ between the angles, and proceed with violence. – 2017-01-03
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    Have you a complete proof? – 2017-01-03
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    Moreover do you have any idea for the second inequality? – 2017-01-03
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    No, sorry, and little time to think about it. :( – 2017-01-04

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