Let $E$ be a vector space over a field $K$. If $u \in \mathscr L(E)$ is an endomorphism of $E$, can it be written as a linear combination of projections (i.e. endomorphisms $p$ of $E$ such that $p \circ p = p$)?


If the dimension of $E$ is finite, this is true. Indeed, one can check that if $(E\_{i,j})\_{1 \leq i,j \leq n}$ is the canonical basis of $\mathscr M\_n(K)$, then $B = (E\_{i,i})\_{1 \leq i \leq n} \cup (E\_{i,i}+E\_{i,j})\_{1 \leq i,j \leq n,\,i \neq j}$ is a basis of $\mathscr M\_n(K)$ and every element of $B$ is a projection matrix.


If $E$ is an Hilbert space and $u$ is bounded, this is also true (see for instance *Fillmore, Peter A. "Sums of operators with square zero." Acta Scientiarum Mathematicarum 28.3-4 (1967): 285*).


In the general case (i.e. arbitrary vector spaces $E$ of infinite dimension), does it still hold ?



Is every endomorphism a linear combination of projections?
The answer is yes: see [Goerge Lowther's answer](https://math.stackexchange.com/a/887623/35400) at Pierre-Yves Gaillard's MathSE linked question: every endomorphism is a $K$-linear combination of $\le 9$ idempotents.


In infinite dimension, the argument even shows that every endomorphism is a $\mathbf{Z}$-linear combination of idempotents (i.e., a sum of $\pm$ idempotents). In finite dimension however this is not true: an endomorphism is a $\mathbf{Z}$-linear combination of idempotents iff its trace belongs to $\mathbf{Z}1\_K$.


