Abstract
The paper concerns linear spaces of functional power series over arbitrary linear spaces with a given bilinear form and involution. The domain of convergence, growth and type or convergence radius resp. of functional power series belonging to certain Hilbert spaces are estimated. Sufficient conditions (concerning growth and type) for a functional to belong to a given Hilbert space are determined. The connection of Hilbert spaces of functional power series and Fock spaces is discussed. The paper contains estimates for functional power series and their derivatives in certain multiplets consisting of topological spaces invariant with respect to differentiation on certain domains. The results are applied to quantum field theory, yielding an estimation of the convergence domain and asymptotic behaviour of the generating functionals for the S -matrix on and off mass shell, for the field and for the derivatives of these quantities with respect to elements of the basic space.
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