Tensor Representation of Spaces of Holomorphic Functions and Applications

In this paper we study the tensor representation of spaces of Fréchet-valued holomorphic functions [H(U,F),τ]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[H(U, F),\tau ]$$\end{document} in the form [(H(U),τ)]⊗^πF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ [(H(U), \tau )] \widehat{\otimes }_{\pi }F$$\end{document} where U is an open subset of a Fréchet space and τ∈{τ0,τω,τδ}.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \in \{\tau _0, \tau _\omega , \tau _\delta \}.$$\end{document} Using this result we consider the following problems: exponential laws for the topologies τ0,τω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _0, \tau _\omega $$\end{document} on the space H(U×V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(U \times V)$$\end{document} where U and V are two open subsets of locally convex spaces E and F respectively; the coincidence of the topologies τ0,τω,τδ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _0, \tau _\omega , \tau _\delta $$\end{document} on spaces of locally convex valued holomorphic functions (resp. germs) H(U, F) [resp. H(K, F)]; the inheritance of the properties (QNo),  (QNo)′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(QNo)'$$\end{document} via the spaces of holomorphic functions and of holomorphic germs.