We investigate the completeness and completions of the normed algebras (D(1)(X), ‖ · ‖) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D(1)(X), ‖ · ‖) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets X for which...

in this paper, the gradual real numbers are considered and the notion of the gradual normed linear space is given. also some topological properties of such spaces are studied, and it is shown that the gradual normed linear space is a locally convex space, in classical sense. so the results in locally convex spaces can be translated in gradual normed linear spaces. finally, we give an examp...