For subspaces, and , of the space, , of all derivatives denotes the set of all such that for all . Subspaces of are defined depending on a parameter . In Section 6, is determined for each of these subspaces and in Section 7, is found for and any of these subspaces. In Section 3, is determined for other spaces of functions on related to continuity and higher order differentiation.
Let C denote the Banach space of real-valued continuous functions on [0,1]. Let Φ: C × C → C. If Φ ∈ +, min, max then Φ is an open mapping but the multiplication Φ = · is not open. For an open ball B(f,r) in C let B²(f,r) = B(f,r)·B(f,r). Then f² ∈ Int B²(f,r) for all r > 0 if and only if either f ≥ 0 on [0,1] or f ≤ 0 on [0,1]. Another result states that Int(B₁·B₂) ≠ ∅ for any two balls B₁ and B₂ in C. We also prove that if Φ ∈ +,·,min,max, then the set is residual whenever E is residual in...
We give a state-of-the-art survey of investigations concerning multivariate polynomial inequalities. A satisfactory theory of such inequalities has been developed due to applications of both the Gabrielov-Hironaka-Łojasiewicz subanalytic geometry and pluripotential methods based on the complex Monge-Ampère operator. Such an approach permits one to study various inequalities for polynomials restricted not only to nice (nonpluripolar) compact subsets of ℝⁿ or ℂⁿ but also their versions for pieces...
We give some necessary and sufficient conditions for an n-1 times differentiable function to be a generalized convex function with respect to an unrestricted n-parameter family.
We prove necessary and sufficient conditions for the validity of the classical chain rule in the Sobolev space and in the space of functions of bounded variation.