Chen, Yuqin; Kim, Jong Kyu Note on finite dimensional \(weak^*\) upper semi-continuous mappings. (English) Zbl 1222.19002 Nonlinear Funct. Anal. Appl. 14, No. 4, 515-519 (2009). First, the authors introduce the following concepts:Let \(E\) be a Banach space and \(E^*\) the dual space of \(E\), and we do not distinguish between the zero in \(E\) and \(E^*\). An operator \(T:D(T)\subseteq E\to 2^{E^*}\) is said to be finite-dimensional weak* upper semi-continuous if for any finite-dimensional subspace \(F\) of \(E\) with \(F\cap D(T)\neq \emptyset\), the restriction \(T:D(T)\cap F\to 2^{E^*}\) is upper-semicontinuous in the weak* topology; \(T\) is said to be finite-dimensional bounded if \(T\) is bounded on any finite-dimensional subspaces.A generalized index for a class of densely defined finite-dimensional weak* upper semi-continuous mappings is constructed. This index is applied to study the range problem of such type of mappings. The basic result is the following:Let \(T:D(T)\subseteq E\to 2^{E^*}\) be a bounded and finite-dimensional weak\(^*\) upper semi-continuous operator with closed convex values, and \(D(T)\) a dense subspace of \(E\). Suppose that \[ \liminf_{x\in D(T),\|x\|\to\infty,f\in Tx}\frac{(f,x)}{\|x\|^2}=+\infty. \]Then \(\overline{T(D(T))}^*=E^*\). Reviewer: Liliana Răileanu (Iaşi) MSC: 19K99 \(K\)-theory and operator algebras 55M10 Dimension theory in algebraic topology 47H11 Degree theory for nonlinear operators Keywords:weak* topology; weak* upper semi-continuous mappings; range problem PDFBibTeX XMLCite \textit{Y. Chen} and \textit{J. K. Kim}, Nonlinear Funct. Anal. Appl. 14, No. 4, 515--519 (2009; Zbl 1222.19002)