Weakly symmetric functions on spaces of Lebesgue integrable functions - Vasylyshyn, T.V.; Zahorodniuk, V.A. (Carpathian Mathematical Publications, 2022)
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 pth power of the absolute value is Lebesgue integrable. We show that every continuous linear functional on L_p[0,1], where p(1,+∞), can be approximated by weakly symmetric continuous linear functionals.