The ‘gap’ is considered to be a meaningful metric on a space of all closed operators defined in a Hilbert space H. The gap between two closed operators S and T is defined as the gap between the graphs  and  of the operators, which are closed subspaces of the product space  Thus, the study of the gap between subspaces has great impact on the gap between operators.

It is quite interesting to note that the gap between two closed subspaces is the norm of the difference of the projection operators on those spaces. We exploit this connection to prove the completeness of the set of closed subspaces under this metric and a few other results.  Also, we establish a result on the gap on Cartesian products.

gap between subspaces, projection operators, filtrations, Cartesian product of spaces.