Let $U, X$ be vector spaces and $f: U \rightarrow X$ be a linear map. Let $X= X_1 \oplus \cdots \oplus X_r$, where $X_i$ is subspace of $X$. Let $\pi_i : X \rightarrow X_i$ be the projection onto $X_i$. How strong a condition is to assume that $\pi_i \circ f (U) \subseteq f(U)$, i.e. that the image of $f$ is $\pi_i$-invariant for all $i=1,...,r$?
I did some thinking in the case where $U=\mathbb{R}, X=\mathbb{R}^2$ and the condition seems quite strong. On the other hand for $U=X=\mathbb{R}$ the condition is trivial.
Any insights?
To the more experienced of our community: how interesting would be a theorem that uses this assumption on the map $f$?
Thanks :-)