List of chapters -- A. General tribes (Boolean lattices) -- A I. Special theorems on Boolean lattices -- B. Important auxiliaries -- B I. General theory of traces -- C. The tribe of figures on the plane -- C I. The trace-theorem -- D. The lattice of subspaces of the Hilbert-Hermite space -- D I. Tribes of spaces -- E. Double scale of spaces -- F. Linear operators permutable with a projector -- G. Some double Stieltjesโ{128}{153} and Radonโ{128}{153}s integrals -- H. Maximal normal operator and its canonical representation -- J. Operators $$N\text{ }\dot{f}\left( {\dot{x}} \right)\text{ = }{}_{df}\phi \text{ }\left( {\dot{x}} \right)\text{ }\cdot \text{ }\dot{f}\left( {\dot{x}} \right)$$ for ordinary functions f -- J I. Operational calculus on general maximal normal operators -- K. Theorems on normal operators and on related canonical mapping -- L. Some classical theorems on normal and selfadjoint operators -- M. Multiplicity of spectrum of maximal normal operators -- N. Some items of operational calculus with application to the resolvent and spectrum of normal operators -- P. Tribe of repartition of functions -- P I. Permutable normal operators -- Q. Approximation of somata by complexes -- Q I. Vector fields on the tribe and their summation -- R. Quasi-vectors and their summation -- R I. Summation of quasi-vectors in the separable and complete Hilbert-Hermite-space -- S. General orthogonal system of coordinates in the separable and complete Hilbert-Hermite-space -- T. Diracโ{128}{153}s Delta-function -- U. Auxiliaries for a deeper study of summation of scalar fields -- W. Upper and lower (DARS)-summation of fields of real numbers in a Boolean tribe in the absence of atoms -- W I. Upper and lower summation in the general case. Complete admissibility. Square summability of fields of numbers -- References -- Alphabetical index