Weakly symmetric functions on spaces of Lebesgue integrable functions

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 Lp[0, 1] of all Lebesgue measur-able 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 Lp[0, 1], where p E (1, +OO), can be approximated by weakly symmetric continuous linear functionals.