Corollary 3

Let E be a separated locally convex space over the field $\mathbf{C}$ and $u$ a compact endomorphism of E. If E is of dimension at least 2, there exists a closed vector subspace F of E, distinct from $\{0\}$ and from E, such that $u(F)\subset F$.

The case $u= 0$ is immediate. The case $u\not = 0$ follows from Cor. 2 by taking $h=u$.