Weakly symmetric functions on spaces of Lebesgue integrable functions

  • T.V. Vasylyshyn Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine
  • V.A. Zahorodniuk Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine
Keywords: symmetric function, weakly symmetric function, holomorphic function on an infinite dimensional space, spaces of Lebesgue integrable functions
Published online: 2022-12-30

Abstract


In this work, we present the notion of a weakly symmetric function. We show that the subset of all weakly symmetric elements of an arbitrary vector space of functions is a vector space. Moreover, the subset of all weakly symmetric elements of some algebra of functions is an algebra. Also we consider weakly symmetric functions on the complex Banach space $L_p[0,1]$ of all Lebesgue measurable complex-valued functions on $[0,1]$ for which the $p$th power of the absolute value is Lebesgue integrable. We show that every continuous linear functional on $L_p[0,1],$ where $p\in (1,+\infty),$ can be approximated by weakly symmetric continuous linear functionals.

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How to Cite
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Vasylyshyn T., Zahorodniuk V. Weakly Symmetric Functions on Spaces of Lebesgue Integrable Functions. Carpathian Math. Publ. 2022, 14 (2), 437-441.

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