Normed algebras of differentiable functions on compact plane setsTools Dales, H.G. and Feinstein, Joel (2010) Normed algebras of differentiable functions on compact plane sets. Indian Journal of Pure and Applied Mathematics, 41 (1). pp. 153187. ISSN 00195588 Full text not available from this repository.AbstractWe investigate the completeness and completions of the normed algebras (D(1)(X),∥•∥) for perfect, compact plane sets X. In particular, we construct a radially selfabsorbing, 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 the completeness of (D(1)(X),∥•∥) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for all starshaped, compact plane sets, and for all Jordan arcs in ℂ.
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