Contributions to the Theories of Topological Groups

Citation

Millsaps, Knox (1944) Contributions to the Theories of Topological Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/bz7q-nh11. https://resolver.caltech.edu/CaltechTHESIS:05292024-220525309

Abstract

The brief abstracts of the papers contained in this thesis are copied from the Bulletin of the American Mathematical Society

Abstract (48-1-96)

An nth order differential is defined for a function F(x) with arguments and values in topological groups and increments in the central subgroup of the argument space with a relativized topology and a generation postulate. The fundamental theorems on unicity, continuity, linear combinations, and iterative functions are then proved.

Abstract (48-5-158)

After an abstract calculus of finite differences is defined, functional definitions of a monomial and polynomial for elements of the group as increments are given. The theorem on the homogeneity of a polynomial is proved for central and arbitrary differences; for central differences the difference being a function of the increment alone implies the difference is a monomial; the independence of the central difference of polynomials and the unique decomposition for the abelian valued case are made to depend on the product of a Vandermonde determinant end a finite product of binomial coefficients. The theory is essentially a generalization of the work of Van der Lijn on abstract polynomials in abelian groups.

Abstract (50-1-48)

By analyzing an example formulated by A. Tychonoff, the spaces H^ρ, o less than or equal to ρ less than ∞, are defined in a manner analogous to that £or classical Hilbert space; some basic properties such as linearity, necessary and sufficient conditions for normability, separability, and sufficient conditions for local convexness are proved.

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