On the Development of Nonlinear Operator Theory
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the Development of Nonlinear Operator Theory Wen Hsiang Wei Received July 7, 2018; in final form, May 28, 2019; accepted August 2, 2019
Abstract. Basic results for nonlinear operators are given. These results include nonlinear versions of the classical uniform boundedness theorem and Hahn–Banach theorem. Further, mappings of a metrizable space into another normed space can belong to some normed spaces if one defines suitable norms. Key words: nonlinear Hahn–Banach theorem, nonlinear operator, nonlinear uniform boundedness theorem. DOI: 10.1134/S0016266320010062
1. Introduction. Operator theory has been at the heart of research in analysis (see [1] and [3; Chap. 4]). Moreover, as implied by [2], considering nonlinear case should be essential. In classical functional analysis, the space of bounded linear operators is a normed space equipped with a reasonable norm. In the next section, a norm function is defined, and some set including certain nonlinear operators from a normed space to a normed space turns out to be a normed space. The Hahn–Banach theorem and the uniform boundedness theorem are fundamental theorems for bounded linear operators between normed spaces. It is meaningful to develop nonlinear counterparts of these theorems. The nonlinear uniform boundedness theorem and the nonlinear Hahn–Banach theorem are given in Secs. 3 and 4, respectively. Some mappings, for example, distributions on some metrizable spaces, do not have “normed” values, because metrizable spaces are not normable. To solve the problem, mappings of a metrizable space into a normed space can have normed values if one defines a suitable norm depending on the metric of the metrizable space. The result for mappings on metrizable spaces is given in the last section. Note that the result for mappings on metrizable spaces can be applied to operators on the space of bounded linear functionals corresponding to the Dirac delta function. In this brief note, only main definitions and theorems along with sketches of the proofs are given. The other results, including other theorems, some corollaries of the theorems, and applications, along with complete proofs of the theorems and corollaries, can be found in the supplementary materials uploaded to arXiv (see [4]). 2. Nonlinear function spaces. Let X and Y be normed spaces over a field K with some norms · X and · Y , respectively, where K is either the real field R or the complex field C. Note that vector spaces and normed spaces in this article are assumed to be nontrivial, i.e., include nonzero elements. Let V (X, Y ) be the set of all operators from X to Y , i.e., the set of arbitrary maps of X into Y . Note that the operators in V (X, Y ) are not assumed to be continuous. An application of algebraic operations to elements F1 , F2 ∈ V (X, Y ) gives the operators F1 + F2 and αF1 from X to Y with (F1 + F2 )(x) = F1 (x) + F2 (x) and (αF1 )(x) = αF1 (x) for x ∈ X, where α ∈ K is a scalar. Further, let the zero element in V (X, Y ) be the operator with image equal to the zero element in Y . The set
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