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The purpose of this book is to ,give the theoretical physicist a geometrical, visual and precise mathematical apparatus which would be better adapted to some of their arguments, than the existing and generally applied methods. The theories, presented in this book, are based on the theory of Boolean lattices, whose elements are closed subspaces in the separable and complete Hilbert-Hermite-space. The first paper, in which the outlines of the said mathematical apparatus is sketched, is that of the author: "Un nouvel appareil mathematique pour la theorie des quanta."] The theory exhibited in this paper has been simplified, generalized and applied to several items of the theory of maximal normal operators in Hilbert-space, especially to the theory of multiplicity of the continuous spectrum and to permutable normal operators, based on a special canonical representation of normal operators and on a general system of coordinates in Hilbert-space, which is well adapted not only to the case of discontinuous spectrum, but also to the continuous one. The normal operators, which can be roughly characterized as operators with orthogonal eigen-vectors and complex eigen-values, constitute a generalization of hermitean selfadjoint and of unitary operators. The importance of the methods, sketched in the mentioned paper, has been emphasized in the review in the "Zentralblatt fUr Mathematik", 2 by the physicist G. LUDWIG and later applied by him in his book "Die Grundlagen der Quantenmechanik"3. The mentioned theory has 1 Annales de l´Institut HENRI POINCARE, tome XI, fasc. II, pages 49-112.

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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 $$N\text{ }\dot{f}\left( {\dot{x}} \right)\text{ = }{}_{df}\phi \text{ }\left( {\dot{x}} \right)\text{ }\cdot \text{ }\dot{f}\left( {\dot{x}} \right)$$ 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. 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.

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