A characterization of the range of a bounded linear transformation in Hilbert space

Abstract:

It is a theorem of Smul’jan and Mac Nerney that for B a bounded linear transformation from a complete complex inner product space $\{S,(\cdot , \cdot )\}$ to S, with adjoint transformation ${B^ \ast },B(S)$ is the set of all z in S for which there is a nonnegative number b such that for all x in $S,|(z,x){|^2} \leqslant b\left \|{B^ \ast }x\right \|{^2}$, in which case if w is that point of ${(\ker B)^ \bot }$ such that $Bw = z$ then the least such b is $\left \|w\right \|{^2}$. This paper provides another description of $B(S)$ and formula for $\left \|w\right \|{^2}$.