Analytic functions in topological vector-spaces

Regular Fréchet-Lie groups of invertible elements in some inverse limits of unital involutive Banach algebras

“…(b) For the corresponding notion of analyticity we refer to [9]. More precisely, let E and F be two Fréchet spaces and let U be a nonempty subset of E; an integer n ≥ 0 being given, a continuous mapping p from E into F is said to be a continuous homogeneous polynomial with degree n if there exists an n-linear mapping p from the Cartesian product…”

“…(c) Section 3 is related to some aspects of differentiability and analyticity on Fréchet spaces according to the ideas of J. Leslie, J. Bochniack, J. Siciak, and some others (see, for example, [8], [9], [6], [10], [5]) and to the notion of generalized Campbell-Baker-Hausdorff Lie group (for shortness: CBH-Lie group) initiated by J. Milnor in [11], which are not necessarily known by nonspecialists in infinite-dimensional analysis and which we shall use in the next sections.…”

Fundamental Problems in the Theory of Infinite-Dimensional Lie Groups

“…β n (y) for all y ∈ Y as a pointwise limit (see [1]). Given real locally convex spaces, following [20] a map f : U → F on an open subset U ⊆ E is called real analytic if it extends to a complex analytic map between open subsets of the complexifications E C and F C .…”

On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

“…(b) If E and F are complex locally convex spaces, then f is called complex analytic if it is continuous and for each x ∈ U there exists a 0-neighborhood V with x + V ⊆ U and continuous homogeneous polynomials (c) If E and F are real locally convex spaces, then we call a map f : [30]). The advantage of this definition, which differs from the one in [5], is that it also works nicely for non-complete spaces. Any analytic map is smooth, and the corresponding chain rule holds without any condition on the underlying spaces, which is the key to the definition of analytic manifolds (see [16] for details).…”