Dilation Groups in Banach, Hilbert, and Euclidean Spaces

This chapter introduces linear dilations in normed vector spaces and studies their properties. The related notions (such as a generator of a dilation) are considered. Some monotonicity properties of dilations in Banach, Hilbert and Euclidean spaces are studied. Geometrical structures induced by linear dilations (such as homogeneous cones, spheres and balls) are presented. A class of linear dilations in function spaces is studied. All results are supported with examples in finite-dimensional and infinite-dimensional spaces.