For each positive integer k, we describe a map f from the complex plane to a suitable non-complete complex locally convex space such that f is k times continuously complex differentiable but not k+1 times, and hence not complex analytic. We also describe a complex analytic map from l^1 to a suitable complete complex locally convex space which is unbounded on each non-empty open subset of l^1. Furthermore, we present a smooth map from the real line to a non-complete locally convex space which is not real analytic although it is given locally by its Taylor series around each point. As a byproduct, we find that free locally convex spaces over subsets of the complex plane with non-empty interior are not Mackey complete.
Examples of differentiable mappings into real or complex topological vector spaces with specific properties are given, which illustrate the differences between differential calculus in the locally convex and the non-locally convex case. In particular, for a suitable non-locally convex space E, we describe a smooth injection of R into E whose derivative vanishes identically; we present a complex C^\infty-map on the complex field C which is not given locally by its Taylor series, around any point; we present a complex C^1-map into a complete, non-locally convex topological vector space which is …
We develop differential calculus of $C^{r,s}$-mappings on products of locally convex spaces and prove exponential laws for such mappings. As an application, we consider differential equations in Banach spaces depending on a parameter in a locally convex space. Under suitable assumptions, the associated flows are mappings of class $C^{r,s}$.
Smooth functions $f:G\to E$ from a topological group $G$ to a locally convex space $E$ were considered by Riss (1953), Boseck, Czichowski and Rudolph (1981), Beltiţă and Nicolae (2015), and others, in varying degrees of generality. The space $C^\infty(G,E)$ of such functions carries a natural topology, the compact-open $C^\infty$-topology. For topological groups $G$ and $H$, we show that $C^\infty(G\times H,E)\cong C^\infty(G,C^\infty(H,E))$ as a locally convex space, whenever both $G$ and $H$ are metrizable or both $G$ and $H$ are locally compact. Likewise, $C^k(G, C^l(H,E))$ can be identifie…
The fundamental problem of calculus of variations is considered when solutions are differentiable curves on locally convex spaces. Such problems admit an extension of the Euler-Lagrange equations [Orlov 2002] for continuously normally differentiable Lagrangians. Here, we formulate a Legendre condition and an extension of the classical theorem of Emmy Noether, thus obtaining first integrals for problems of the calculus of variations on locally convex spaces.
Usually, for extension of local maps, one uses multiplication by so called bump functions. However, majority of infinite-dimensional linear topological spaces do not have smooth bump functions. Therefore, in \cite{BR} we suggested a new approach for Banach spaces, based on the composition with locally identical maps. In the present work we discuss a possibility of generalization of this method for arbitrary spaces and applications of this theory.
In this work, various versions of the so-called Omega-Lemma are provided, which ensure differentiability properties of pushforwrds between spaces of C^r-sections (or compactly supported C^r-sections) in vector bundles over finite-dimensional base manifolds whose fibres are (possibly infinite-dimensional) locally convex spaces. Applications are given, including the proof of continuity for some natural module multiplications on spaces of sections and the construction of certain infinite-dimensional Lie groups of Lie group-valued maps.
In this paper, we introduce restricted products for families of locally convex spaces and formulate criteria ensuring that mappings into such products are continuous or smooth. As a special case, can define restricted products of weighted function spaces and obtain results concerning continuity and differentiability properties of natural non-linear mappings between such spaces. These concepts and results are the basis for the study of weighted vector fields on Riemannian manifolds in a subsequent work (see [B. Walter, "Weighted diffeomorphism groups of Riemannian manifolds", arXiv: 1601.02834]…
We prove a version of the Stokes formula for differential forms on locally convex spaces. The main tool used for proving this formula is the surface layer theorem proved in another paper by the author. Moreover, for differential forms of a Sobolev-type class relative to a differentiable measure, we compute the operator adjoint to the exterior differential in terms of standard operations of calculus of differential forms and the logarithmic derivative. Previously, this connection was established under essentially stronger assumptions.
We study spaces $\mathcal{CV}^{k}(Ω,E)$ of $k$-times continuously partially differentiable functions on an open set $Ω\subset\mathbb{R}^{d}$ with values in a locally convex Hausdorff space $E$. The space $\mathcal{CV}^{k}(Ω,E)$ is given a weighted topology generated by a family of weights $\mathcal{V}^{k}$. For the space $\mathcal{CV}^{k}(Ω,E)$ and its subspace $\mathcal{CV}^{k}_{0}(Ω,E)$ of functions that vanish at infinity in the weighted topology we try to answer the question whether their elements can be approximated by functions with values in a finite dimensional subspace. We derive suff…
In this paper we study the problem of extending functions with values in a locally convex Hausdorff space $E$ over a field $\mathbb{K}$, which have weak extensions in a weighted Banach space $\mathcal{F}ν(Ω,\mathbb{K})$ of scalar-valued functions on a set $Ω$, to functions in a vector-valued counterpart $\mathcal{F}ν(Ω,E)$ of $\mathcal{F}ν(Ω,\mathbb{K})$. Our findings rely on a description of vector-valued functions as linear continuous operators and extend results of Frerick, Jordá and Wengenroth. As an application we derive weak-strong principles for continuously partially differentiable fun…
We show that if $X$ is a Banach space whose dual $X^{*}$ has an equivalent locally uniformly rotund (LUR) norm, then for every open convex $U\subseteq X$, for every $\varepsilon >0$, and for every continuous and convex function $f:U \rightarrow \mathbb{R}$ (not necessarily bounded on bounded sets) there exists a convex function $g:X \rightarrow \mathbb{R}$ of class $C^1(U)$ such that $f-\varepsilon\leq g\leq f$ on $U.$ We also show how the problem of global approximation of continuous (not necessarily bounded on bounded sets) and convex functions by $C^k$ smooth convex functions can be reduced…
We establish exponential laws for certain spaces of differentiable functions over a valued field K. For example, we show that the topological vector spaces C^{r,s}(U x V,E) and C^r(U,C^s(V,E)) are isomorphic if U and V are open subsets of K^n and K^m, respectively, E is a topological K-vector space, and r,s are degrees of differentiability. As a first application, we study the density of locally polynomial functions in spaces of partially differentiable functions over an ultrametric field (thus solving an open problem by E. Nagel), and also global approximations by polynomial functions. As a s…
We investigate metric properties of level sets of horizontally differentiable maps defined on the first Heisenberg group $(\Bbb{H}^1,d_{cc})$ equipped with the standard sub-Riemannian structure. In particular, we present an exhaustive analysis in a new case of a map $F\in C^1_H(\Bbb{H}^1, \Bbb{R}^2)$ with surjective horizontal differential (an analogue of the classical implicit function theorem). Among other results, we show that a level set of such map is locally a simple curve of Hausdorff sub-Riemannian dimension 2, but, surprisingly, in general its two-dimensional Hausdorff measure can be …
Operations on $\mathscr{M}\otimes_{\mathcal{A}(U)}\mathrm{Alt}^{\bullet}(U,M)$. Stokes theorem for locally convex linear space valued forms. Divergence theorem for locally convex linear space valued vector fields
We prove that functions of locally bounded deformation on $\mathbb{R}^n$ are $\mathrm{L}^{n/(n-1)}$-differentiable almost everywhere. More generally, we show that this critical $\mathrm{L}^p$-differentiability result holds for functions of locally bounded $\mathbb{A}$-variation, provided that the first order, homogeneous, linear differential operator $\mathbb{A}$ has finite dimensional null-space.
In this work we establish a Stokes-type integral equality for scalarly essentially integrable forms on an orientable smooth manifold with values in the locally convex linear space $\langle B(G),σ(B(G),\mathcal{N})\rangle$, where $G$ is a complex Banach space and $\mathcal{N}$ is a suitable linear subspace of the norm dual of $B(G)$. This result widely extends the Newton-Leibnitz-type equality stated in one of our previous articles. To obtain our equality we generalize the main result of that article, and employ the Stokes theorem for smooth locally convex vector valued forms established in a p…
Let $\mathcal{N}$ be a non-Archimedean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD$^k$) functions, and WLUD$^\infty$ functions at a point or on an open subset of $\mathcal{N}$. Then we show under what conditions a WLUD$^\infty$ function at a point $x_0\in\mathcal{N}$ is analytic in an interval around $x_0$, that is, it h…
We examine the theory of metric currents of Ambrosio and Kirchheim in the setting of spaces admitting differentiable structures in the sense of Cheeger and Keith. We prove that metric forms which vanish in the sense of Cheeger on a set must also vanish when paired with currents concentrated along that set. From this we deduce a generalization of the chain rule, and show that currents of absolutely continuous mass are given by integration against measurable $k$-vector fields. We further prove that if the underlying metric space is a Carnot group with its Carnot-Carathéodory distance, then every…
This article is devoted to the investigation of wrap groups of connected fiber bundles over the fields of real $\bf R$, complex $\bf C$ numbers, the quaternion skew field $\bf H$ and the octonion algebra $\bf O$. These groups are constructed with mild conditions on fibers. Their examples are given. It is shown, that these groups exist and for differentiable fibers have the infinite dimensional Lie groups structure, that is, they are continuous or differentiable manifolds and the composition $(f,g)\mapsto f^{-1}g$ is continuous or differentiable depending on a class of smoothness of groups. Mor…
We shall prove that a moduli space of flat irreducible Lie algebroid connections over a compact manifold has locally a natural structure of a smooth differentiable space. This is a generalization of some well known results for the moduli space of holomorphic structures on a complex vector bundle over a compact complex manifold.
In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations $π\: G \to \GL(V)$ of an infinite dimensional Lie group $G$ on a locally convex space $V$. The first class of results concerns the space $V^\infty$ of smooth vectors. If $G$ is a Banach--Lie group, we define a topology on the space $V^\infty$ of smooth vectors for which the action of $G$ on this space is smooth. If $V$ is a Banach space, then $V^\infty$ is a Fréchet space. This applies in particular to $C^*$-dynamical systems $(\cA,G, α)$, where $G$ is a Banach--Lie gr…
We prove two theorems about differentiable functions on the Banach space C(K), where K is compact. (i) If C(K) admits a non-trivial function of class C^m and of bounded support, then all continuous real-valued functions on C(K) may be uniformly approximated by functions of class C^m. (ii) If C(K) admits an equivalent norm with locally uniformly convex dual norm, then C(K) admits an equivalent norm which is of class C^infty (except at 0).
Following Malý's definition of absolutely continuous functions of several variables, we consider $Q$-absolutely continuous mappings $f\colon X\to V$ between a doubling metric measure space $X$ and a Banach space $V$. The relation between these mappings and Sobolev mappings $f\in N^{1,p}(X;V)$ for $p\ge Q$ is investigated. In particular, a locally $Q$-absolutely continuous mapping on an Ahlfors $Q$-regular space is a continuous mapping in $N^{1,Q}_{\rm{loc}}(X;V)$, as well as differentiable almost everywhere in terms of Cheeger derivatives provided $V$ satisfies the Radon-Nikodym property. Conv…
