Modeling Spheres in Some Paranormed Sequence Spaces

Abstract

We introduce a new sequence space hA(p), which is not normable, in general, and show that it is a paranormed space. Here, A and p denote an infinite matrix and a sequence of positive numbers. In the special case, when A is a diagonal matrix with a sequence d of positive terms on its diagonal and p=(1,1,⋯), then hA(p) reduces to the generalized Hahn space hd. We applied our own software to visualize the shapes of parts of spheres in three-dimensional space endowed with the relative paranorm of hA(p), when A is an upper triangle. For this, we developed a parametric representation of these spheres and solved the visibility and contour (silhouette) problems. Finally, we demonstrate the effects of the change of the entries of the upper triangle A and the terms of the sequence p on the shape of the spheres.

On the New Generalized Hahn Sequence Space h d p

“…Besides, the authors stated and proved various significant results concerning characterization of matrix transformations between the space h d and classical BK spaces, and characterization of compact operators on the space h d using Hausdorff measure of noncompactness. We refer to [2,10,[13][14][15][16][17][18][19][20][21][22][23] and the survey paper [22] for more studies and results related to Hahn sequence space.…”