The notion of functions dependent locally on finitely many coordinates plays an important role in the theory of smoothness and renormings on Banach spaces, especially when higher order smoothness is involved. In this note we investigate the structural properties of Banach spaces admitting (arbitrary) bump functions depending locally on finitely many coordinates.
Let be a completely regular Hausdorff space and a real normed space. We examine the general properties of locally solid topologies on the space of all -valued continuous and bounded functions from into . The mutual relationship between locally solid topologies on and
Proporcionamos una demostración muy corta de un teorema de Cascales-Orihuela que establece que todo conjunto precompacto de un espacio localmente convexo de la clase G (en el sentido de Cascales-Orihuela) es metrizable.