Please, I need help with this proble.

Let $(H,\langle\cdot,\cdot\rangle)$ be a Hilbert space and let $V_1,V_2,\ldots,V_N$ closed subspaces, mutually orthogonal of $H$, that is, $v_i\perp v_j$ $\forall v_i\in V_i$, $\forall v_j\in V_j$, $\forall i\neq j$. Show that $$\mathbb{I} - \mathbb{P}\ =\ \mathbb{P}_1 + \mathbb{P}_2 + \ldots + \mathbb{P}_N,$$ where $\mathbb{P}:H\rightarrow V_1^{\perp}\cap V_2^{\perp}\cap\ldots\cap V_N^{\perp}\;$ and $\;\mathbb{P}_j:H\rightarrow V_j$ are the orthogonal projectors, respectively.

Thanx so much in advance.


Let $x\in H$ then we have $$x=\mathbb{P}_i(x)+(x-\mathbb{P}_i(x))\quad i=1,\ldots N$$ so $$x-(x-\sum_{i=1}^N \mathbb{P}_i(x))=\sum_{i=1}^N \mathbb{P}_i(x)$$

Moreover it's easy to see that $$(\mathbb{I}-\sum_{i=1}^N\mathbb{P}_i)(\mathbb{I}-\sum_{i=1}^N\mathbb{P}_i)=(\mathbb{I}-\sum_{i=1}^N\mathbb{P}_i)$$ hence $\displaystyle (\mathbb{I}-\sum_{i=1}^N\mathbb{P}_i)$ is a projection and $$\mathbb{P}_i(\mathbb{I}-\sum_{i=1}^N\mathbb{P}_i)=0\quad,\quad (\mathbb{I}-\sum_{i=1}^N\mathbb{P}_i)(x)=0\quad\forall x\in V_i$$ so $(\mathbb{I}-\sum_{i=1}^N\mathbb{P}_i)$ is the orthogonal projection onto $ V_1^{\perp}\cap V_2^{\perp}\cap\ldots\cap V_N^{\perp}$ and parallel to $\sum_{i=1}^N V_i$

and this prove that $$\mathbb{P}(x)=(x-\sum_{i=1}^N \mathbb{P}_i(x))$$

finally we find $$x-\mathbb{P}(x)=\sum_{i=1}^N \mathbb{P}_i(x)$$


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