Cq-COMMUTING MAPS AND INVARIANT APPROXIMATIONS

Let X be a linear space. A p-norm on X is a real-valued function on X with 0 < p ≤ 1, satisfying the following conditions: (i) ‖x‖p ≥ 0 and ‖x‖p = 0⇔ x = 0, (ii) ‖αx‖p = |α|p‖x‖p, (iii) ‖x+ y‖p ≤ ‖x‖p +‖y‖p, for all x, y ∈ X and all scalars α. The pair (X ,‖ · ‖p) is called a p-normed space. It is a metric linear space with a translation invariant metric dp defined by dp(x, y)= ‖x− y‖p for all x, y ∈ X . If p = 1, we obtain the concept of the usual normed space. It is well known that the topology of every Hausdorff locally bounded topological linear space is given by some p-norm, 0 < p ≤ 1 (see [7, 13] and references therein). The spaces lp and Lp, 0 < p ≤ 1, are p-normed spaces. A p-normed space is not necessarily a locally convex space. Recall that dual space X∗ (the dual of X) separates points of X if for each nonzero x ∈ X , there exists f ∈ X∗ such that f (x) = 0. In this case the weak topology on X is well defined and is Hausdorff. Notice that if X is not locally convex space, then X∗ need not separate the points of X . For example, if X = Lp[0,1], 0 < p < 1, then X∗ = {0} [17, pages 36–37]. However, there are some nonlocally convex spaces X (such as the pnormed spaces lp, 0 < p < 1) whose dual X∗ separates the points of X . In the sequel, we will assume that X∗ separates points of a p-normed space X whenever weak topology is under consideration.