Remarks on weakly continuous functions in banach spaces

Abstract

Let E be a Banach space over the real.s and let E* be the dual space. Le t = (∝1 , …, ∝n) be a finite sequence of non-negative integers and u = (u1, …,un) a finite sequence of elements in E*. The notation u∝ = u1 u1∝1 … u1∝1 … un∝n is standard and will used throughout. We will write |∝| = ∝1 + … + ∝n . Any real valued function in E of the form P = ∑ (|∝|≤n) a∝ u∝ , a∝ a real number, is said to be a polynomial. Clearly, every polynomial is weakly continuous.