Remarks on weakly continuous functions in banach spaces

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, …,u_n) a finite sequence of elements in E*. The notation u^∝ = u₁ u₁^∝₁  … u₁^∝₁ … u_n^∝_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.