The Mathematical Apparatus for Quantum-Theories

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 Stieltjes' and Radon's integrals.- H. Maximal normal operator and its canonical representation.- J. Operators $$Ntext{ }dot{f}left( {dot{x}} right)text{ = }{}_{df}phi text{ }left( {dot{x}} right)text{ }cdot text{ }dot{f}left( {dot{x}} right)$$ 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. Dirac'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.