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I am reading, the book Manifolds, Tensor analysis of Abraham, Marsden and Ratiu. In particular, there are several points that I do not understand about a Grassmann manifold. The example starts like this,enter image description here

I can not see why the applications $\pi_{G}(F,H)$ is in $L(F,H))$ and why is invertible. (Maybe, it's because I do not understand what the projections are doing).

Later in the demonstration, they define for $\alpha\in L(F,G)$ the graph of $\alpha$ by $$\Gamma_{F,G}(\alpha)=\{f+\alpha(f);f\in F\}$$ which is a closed subspace of $E=F\oplus G$, but I can not see this either, I thought to occupy the theorem of the closed graph.

Could you please help me, thank you!

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Let $x\in F$, suppose that $\pi_G(F,H)(x)=0$, you can write $x=x_G+x_H$ where $x_G\in G$ and $x_H\in H$, you deduce that $\pi_G(F,H)(x)=\pi_H(x)=x_H=0$, this implies that $x\in G$, since $G\cap F=0$, you deduce that $x=0$.

Let $y\in H$ write $y=x_F+x_G, x_F\in F, x_G\in G, \pi_G(F,H)(y)=\pi_H(x_F+x_G)=\pi_H(x_F)=y$ implies that $\pi_G(F,H)$ is surjective. You deduce that $\pi_G(F,H)$ is bijective. The open mapping theorem implies that $\pi_G(F,H)$ is invertible.

For the second question, consider: $L:F\oplus G\rightarrow G$ defined by $L(f+g)=g-\alpha(f))$, show that $L$ is continuous and consider $L^{-1}(0)$.

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