A GROUP INVARIANT BISHOP-PHELPS THEOREM

We show that for any Banach space and any compact topological group G subset of L(X) such that the norm of X is G-invariant, the set of norm attaining G-invariant functionals on X is dense in the set of all G-invariant functionals on X, where a mapping f is called G-invariant if for every x is an element of X and every g E G, f(g(x)) = f (x). In contrast, we show also that the analog of Bollobas result does not hold in general. A version of Bollobas and James' theorems is also presented.