Gâteaux derivatives and their applications to approximation in Lorentz spaces Γp,w

We establish the formulas of the left- and right-hand Gâteaux derivatives in the Lorentz spaces Γp,w = {f: ∫α0(f**)Pw\u3c ∞}, where 1 ≤ p \u3c ∞, w is a nonnegative locally integrable weight function and f** is a maximal function of the decreasing rearrangement f* of a measurable function f on (0,α), 0 \u3c α ≤ ∞. We also find a general form of any supporting functional for each function from Γp,w, and the necessary and sufficient conditions for which a spherical element of Γp,w is a smooth point of the unit ball in Γp,w. We show that strict convexity of the Lorentz spaces Γp,w is equivalent to 1 \u3c p \u3c ∞ and to the condition ∫0∞w = ∞. Finally we apply the obtained characterizations to studies the best approximation elements for each function f ∈ Γp,w from any convex set K ⊂ Γp,w. © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim