Abstract. Recently, using perturbation theory, A. Dijksma and A. Gheondea proved the relation $$\kappa ^+(L^\bot )+{\dim }(L\cap H^-)=\kappa ^-(L) +{\dim }(L^\bot\cap H^+),$$ where $H^+\oplus H^-$ is a fundamental decomposition of a Kreĭn space $H$; $L$ is a closed subspace of $H$; $\kappa ^+$ and $\kappa ^-$ denote positive and negative signature respectively, and no distinction between infinite cardinals is made. We give an elementary proof of this relation.

AMS Subject Classification (1991): 46C20

Received August 11, 1994. (Registered under 5692/2009.)