On Point Values of Boehmians
V. Karunakaran, Ramachandran Vembu · Rocky Mountain Journal of Mathematics · 2005
The notion of a value of a Boehmian at a point, its properties and the concept of regular delta sequences are available in the literature.Let E be a Banach space.Denote by C(R N , E) the space of all continuous E-valued functions on R N and by D(R N ) the space of all infinitely differentiable real-valued functions with compact support in R N .Using C(R N , E) as the top space and the usual delta sequences from D(R N ) we can construct in a canonical way a Boehmian space B = B(R N , E).In 1994, Piotr Mikusiński and Mourad Tighiouart asserted that, if for every representation [f n /φ n ] of F ∈ B where (φ n ) is regular delta sequence we have lim n→∞ f n (x 0 ) = a, then F (x 0 ) = a.In this paper we shall point out that the proof of this theorem contains an error, produce a counterexample to show that the theorem is not valid and obtain modified conditions for its validity.As a consequence we shall also show that if F =[f n /φ n ] where (φ n ) is a delta sequence made of one function and if lim n→∞ f n (x 0 ) = a for every such representation, then F need not have a value at x 0 .Incidentally, this observation settles one of the questions raised by Piotr Mikusiński and Mourad Tighiouart.