On near-standardness in Hilbert spaces

The notions of nearstandardness and shadow are generalizations of convergence and limit respectively. There exist many useful variants of these notions. Here we collect some special aspects and properties of shadows and nearstandardness. This paper concerns the following: the shadows of a vector and an operator, weak, strong, and uniform nearstandardness, use of the Hilbert–Schmidt norm, properties of the map $A\mapsto{}^o{A}$, nearstandard projections and subspaces, conditions for graph-nearstandardness, and examples. We work with IST, i.e. Internal Set Theory, a version of nonstandard analysis suggested by Edward Nelson