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

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