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 and lt;x, u and gt; 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 and lt;x, P* u and gt; = and lt;P x, u and g...