A characterization of (locally) uniformly convex spaces in terms of relative openess of quotient maps on the unit ball
Let \(X\) be a real normed linear space and \(U\) its closed unit ball. Definition. Let \(\tau\) be a family of maps defined on \(X\) and let \(A\) be a subset of \(X\). The maps from \(\tau\) are said to be equally relatively open on \(A\) if for each \(x\in A\) and \(\varepsilon> 0\) there is a \(\rho> 0\) such that every \(T\in\tau\) maps the \(\varepsilon\)-neighbourhood of \(x\) in \(A\) onto a set containing the \(\rho\)-neighbourhood of \(T(x)\) in \(T(A)\). If such a \(\rho\) can be chosen so that it depends only on \(\varepsilon\) (and not on \(x\)), the maps from \(\tau\) are said to be uniformly relatively open on \(A\). For any closed linear subspace \(M\) of \(X\), let \(Q_M\) denote the quotient map associated with \(M\), \(Q_M:X\to X/M\). In this paper the author studies quantitatively the relative openness of quotient maps on \(U\). Indicative results in the paper are: Theorem 1. The following statements are equivalent: (i) \(X\) is locally uniformly convex. (ii) The quotient maps \(Q_M:X\to X/M\) associated with the closed linear subspaces \(M\) of \(X\) are equally relatively open on \(U\). (iii) For any family of linear maps defined on \(X\), equal relative openness on \(X\) implies equal relative openness on \(U\). Theorem 2. The following statements are equivalent: (i) \(X\) is uniformly convex. (ii) The quotient maps \(Q_M:X\to X/M\) associated with the closed linear subspaces \(M\) of \(X\) are equally uniformly relatively open on \(U\). (iii) For any family of linear maps defined on \(X\), equal relative openness on \(X\) implies equal uniform relative openness on \(U\).
scientific article; zbMATH DE number 1542412
scientific article