Can anyone give me $k$ projection maps $P_i:V \to V$ ;$i=1,...,k$ such that $\sum_{i=1}^k Im(P_i)$ is not a direct sum but $\sum_{i=1}^k P_i=Id$. Can anyone give me $k$ projection maps $P_i:V \to V$ ;$i=1,...,k$ such that $\sum_{i=1}^k Im(P_i)$ is not a direct sum but $\sum_{i=1}^k P_i=Id$.
I think in $\Bbb R^2$ I can't get but in $\Bbb R^2$ I am not getting any example right now and difficult to see.

CLARIFICATION : This answers the case of finite-dimensional vector spaces over $\mathbb R$ (or any field with zero characteristic).
Such a counterexample does not exist. Indeed, let $V_i={\textsf{Im}}(P_i)$ and $r_i={\textsf{dim}}(V_i)={\textsf{trace}}(p_i)$ since $p_i$ is a projector. Since $\sum_{i=1}^k p_k=\textsf{Id}$, we have
$x=\sum_{i=1}^k px$ for any $x\in V$, whence $V=\sum_{i=1}^k V_i$ and in particular $$\textsf{dim}(V) \leq \sum_{i=1}^k \textsf{dim}(V_i)=\sum_{i=1}^k r_i\tag{1}$$.
But taking the traces in $\sum_{i=1}^k p_k=\textsf{Id}$, we see that (1) must in fact be an equality, which forces the sum to be direct.
