Linear topological properties of the space of analytic functions on the real line (Q2760122)

The two authors recently proved the striking theorem that, for a nonvoid open subset \(\Omega\subset \mathbb{R}^N\), the space \(A(\Omega)\) of all real analytic functions on \(\Omega\) does not have a (Schauder) basis [see Stud. Math. 142, 187-200 (2000; Zbl 0990.46015)]. In particular, \(A(\Omega)\) is the first ``natural'' example of a separable function space without a basis: \(A( \Omega)\) is a nuclear locally convex space with good properties which has occurred in important applications and was not just constructed with the purpose of obtaining a space without a basis. Incidentally, while \(A(\Omega)\) certainly has the approximation property, it seems to be unknown whether it has the bounded approximation property. -- In the article under review, the authors concentrate on the one dimensional case, i.e., on subsets \(\omega\) of \(\mathbb{R}\). This allows for proofs which are elementary and do not require deeper tools from complex analysis in several variables. On the other hand, the functional analytic tools remain the same and are just reported, without proofs.NEWLINENEWLINENEWLINEIn the first part, the locally convex properties of \(A(\omega)\) are surveyed. Two natural locally convex topologies coincide on \(A(\omega)\), a result originally due to \textit{A. Martineau} [Math. Ann. 163, 62-88 (1966; Zbl 0138.38101)]. \(A(\omega)\) is a complete ultrabornological (PLN)-space in which the polynomials are dense. The strong dual of \(A(\mathbb{R})\) has a representation (via Fourier-Laplace transform and the Paley-Wiener theorem) as a weighted (LF)-space of entire functions. Next, special emphasis is given to \((\Omega)\)-type conditions for \(A(\mathbb{R})\), a vector valued splitting lemma and their consequences: Every complemented Fréchet subspace of \(A(\mathbb{R})\) is finite dimensional, (*) every complemented subspace of \(A(\mathbb{R})\) with a basis is a (DF)-space. In addition, an interesting interpolation result holds. In the last section, convolution operators \(T_\mu\) on \(A(I)\) for an open interval \(I\subset\mathbb{R}\) are treated. Using (*), it is characterized for \(\text{supp} \mu= \{0\}\) when \(T_\mu\) has a continuous linear right inverse on \(A(I)\). For results of this type in the general case, see \textit{J. Bonet}, \textit{P. Domański} and \textit{D. Vogt} [J. Lond. Math. Soc., II. Ser. 66, 407-420 (2000; Zbl 1027.46048)] and \textit{M. Langenbruch} [Stud. Math. 110, 65-82 (1994; Zbl 0824.35147)].NEWLINENEWLINEFor the entire collection see [Zbl 0971.00063].