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 itself. 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.