On the convergence of galerkin approximations

Let X be a separable Banach space over the reals and let X· be its dual. If x ϵ  X and u ϵ X* we will write <x, u> instead of u (x).  Also, if P: X → X is a linear operator we will denote by  P* the adjoint from X* into X* which is defined by <x, P* u > = <P x, u>. The strong convergence in X will be denoted by x_n → x, the weak convergence in X by x_n^w → x and the w* - convergence in X* by  x_n^w*→ x.

We sat that X has property (B) if there is a sequence { P_n } of bounded linear operator from X into itself such that,