A Type of Pseudo-Norm

Citation

Diamond, Robert James (1951) A Type of Pseudo-Norm. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/EDZW-V217. https://resolver.caltech.edu/CaltechTHESIS:10132017-091346134

Abstract

In mathematical literature, the term pseudo-norm has no one specific definition but is used for functionals satisfying some but not all of the postulates for a norm. The notion of such functionals or "pseudo-norms" is common in the study of linear topological spaces,¹⁾ which, from one point of view, may be regarded as generalizations of normed linear spaces. The particular type of pseudo-norm considered in this thesis is the triangular norm of Menger's "generalized vector space".²⁾ Menger noticed that only the triangle property of the norm was necessary in order to obtain certain results in the calculus of variations, and thought that a linear space with a generalized triangular "distance" might prove to be a fruitful concept.

We first consider spaces (type K, see text) which are more specialized than those treated by Menger. In this thesis, spaces of the latter type are termed "spaces of type G". Apart from the intrinsic interest of type K spaces, certain aspects of their theory are applied in Chapter IV to the treatment of spaces of type G.

In Chapter I a space of type K is defined, the independence of the pseudo-norm postulates is established, and the question of the continuity of the pseudo-norm is treated. In Chapter II the notion of equivalence classes leads to a vector space of type K/Z, the existence of which depends only on the presence of a pseudo-norm in K. The more general spaces of type G are then introduced. A metric topology defined in terms of the pseudo-norm is discussed in Chapter III and functionals linear with respect to this topology are considered.

The question of the Gâteaux differentiability of the pseudo-norm is taken up in Chapter IV and a connection is established between this property and the existence of functionals linear in the topology of the pseudo-norm.

Chapter V investigates connections between the pseudo-norm and ordering relations in a real vector space. Conditions are found under which a partial ordering can be defined in terms of a given pseudo-norm, and conversely.

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1). See, for example, Hyers, Ref. 7; LaSalle, Ref. 9; von Neumann, Ref. 16; Wehausen, Ref. 19.

2). Menger, Ref. 13, p 96.

Thesis Files