On the continuous linear operator in abstract analytic spaces (Q2730534)

The author considers the locally convex abstract analytic space either as the projective or as the inductive limit of a sequence of Banach spaces. Let \(\omega_{K}\) denote the set of all multisequences \(x=(x_{k}), k\in K\), of complex numbers \(x:K\to\mathbb{C}\), where \(K\) is an infinite subset of \(\mathbb{Z}^{n}\). Let \(\Phi\) be a homogeneous characteristic. Consider the abstract analytic space \({\mathcal A}_{\Phi}(\varphi)=\{x\in\omega_{K}:\chi_{\Phi}^{x}\leq\varphi\}\), where function \(\varphi:\Sigma_{K}\to \mathbb{R}_{+}\) is upper semicontinuous. The author constructs the sequence of Banach spaces \(B_1\supset B_{2}\supset\ldots\) such that \({\mathcal A}_{\Phi}(\varphi)\) equipped with the strong topology is the projective limit of spaces \(B_{j}: {\mathcal A}_{\Phi}(\varphi)=\bigcap_{j\in \mathbb{N}}B_{j}\). Similar results are obtained in the case of lower semicontinuous function \(\varphi\), but the considered limit is inductive. The general form of continuous linear operator in locally convex abstract analytic spaces is described in terms of the infinite multidimensional matrices.