Banach–Stone Theorem for Quaternion- Valued Continuous Function Spaces

We prove some Banach–Stone type theorems for linear isometries of vector-valued continuous function spaces, by making use of the extreme point method.

In this paper we embark on the study of Dynamic Systems of Shifts in the space of piece-wise continuous functions analogue to the known Bebutov system. We give a formal definition of a topological dynamic system in the space of piece-wise continuous functions and show, by way of an example, stability in the sense of Poisson discontinuous function. We prove that a fixed discontinuous function, f, is discontinuous for all its shifts, whereas the trajectory of discontinuous function is not a compact set.

1. In this paper, we consider the problem ofmultiplication of a generalized function by a discontinuous function. The need in such an operation appears in the study of ordinary differential equations with generalized functions [1–8]; it can be viewed as a particular case of a more general operation of multiplication of two generalized functions [9–11] the latter operation arises, in particular, in applications to the problems of quantum mechanics [12, 13]. As is known [14, 15], in the classical space D′, it is impossible to define a continuous operation of multiplication of generalized functions. It is also impossible to define a continuous partial operation of multiplication of a generalized function from D′ by a discontinuous function [14, 15]. A continuous operation of elements of D′ is defined in the algebra of Colombeau generalized functions [9], which contains D′ as a subspace. In the algebra of Colombeau generalized functions, there exists a product of any two elements of D′, although, in the general case, this product is a Colombeau generalized function and does not belong to D′ [9]. In particular, the product of the unit function θτ discontinuous at τ and the delta-function δτ does not belong to D′. This leads to a series of inconveniences in the study of differential equations with generalized functions containing the product of a generalized function and a discontinuous function. The main inconvenience is in the fact that the solution is not an ordinary discontinuous function but a Colombeau generalized function, which makes it difficult to give its physical interpretation [9, 16]. In this paper, we construct a space of dynamical generalized functions T ′ in which a continuous and associative operation of multiplication of a generalized function by a discontinuous function is defined. The definitions of the product of the unit function θτ and the classical delta-function δτ ∈ D′ given in [4, 11, 17, 18] (which, generally speaking, are neither associative, nor continuous) are particular cases of the multiplication in the space T ′.

This paper is mainly concerned with the study of the generators of those positive -semigroups on weighted continuous function spaces that leave invariant a given closed sublattice of bounded continuous functions and whose relevant restrictions are Feller semigroups. Additive and multiplicative perturbation results for this class of generators are also established. Finally, some applications concerning multiplicative perturbations of the Laplacian on , , and degenerate second-order differential operators on unbounded real intervals are showed.

Quasilinear evolutionary equations and continuous interpolation spaces

Many inverse problems involve the estimation of a high dimensional quantity, such as a function, from noisy indirect measurements. These problems have received much study from both classical and statistical directions, with each approach having its own advantages and disadvantages. In this thesis we focus on the Bayesian approach, in which all uncertainty is modelled probabilistically. Recently the Bayesian approach to inversion has been developed in function space. Much of the existing work in the area has been focused on the case when the prior distribution produces samples which are continuous functions, however it is of interest, both in terms of applications and mathematically, to consider cases when these samples are discontinuous. Natural applications are those in which we wish to infer the shape and locations of interfaces between different materials, such as in tomography. In this thesis we consider Bayesian inverse problems in which the unknown function is piecewise continuous or piecewise constant. Based on prior information, the problem is then to infer the discontinuity set, the values the function takes away from the discontinuities, or both simultaneously. These problems are considered both from analytic and computational points of view. In order to ensure numerical robustness, we formulate any algorithms directly on function space before discretizing. This requires a number of technical issues to be considered, such as the equivalence and singularity of measures on such spaces.

Given two subsets X and Y of ℝ each with at least two points, we describe the surjective linear isometries between the spaces of functions of bounded variation BV(X) and BV(Y): namely, if T : BV(X) → BV(Y) is such an isometry, then there exist α ∈ ℂ, |α| = 1, and a monotonic bijective map h : Y → X such that (Tf)(y) = αf(h(y)) for every f ∈ BV(X) and every y ∈ Y.

For any compact Hausdorff space K and any Banach space K , let C(K, X) be the space of all continuous X-valued functions on K with the norm $$ \left\| f \right\| = \mathop {\sup }\limits_{t \in K} \left\| {f(t)} \right\|_X . $$

Surjective isometries between some classical function spaces are investigated. We give a simple technical scheme which verifies whether any such isometry is given by a homeomorphism between corresponding Hausdorff compact spaces. In particular the answer is positive for the C 1 ( X ) {C^1}(X) , AC ⁡ [ 0 , 1 ] \operatorname {AC} [0,1] , Lip α ( X ) {\operatorname {Lip} _\alpha }(X) and lip α ( X ) {\operatorname {lip} _\alpha }(X) spaces provided with various natural norms.

Assume both X X and Y Y are Riemann surfaces which are subsets of compact Riemann surfaces X 1 X_1 and Y 1 , Y_1, respectively, and that the set X 1 − X X_1 - X has infinitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic differentials on X X and Y Y are the ones induced by conformal homeomorphisms and complex constants of modulus 1. It follows that every biholomorphic map from the Teichmüller space of X X onto the Teichmüller space of Y Y is induced by some quasiconformal map of X X onto Y Y . Consequently we can find an uncountable set of Riemann surfaces whose Teichmüller spaces are not biholomorphically equivalent.

Equivalence of minimax and viscosity solutions of path-dependent Hamilton–Jacobi equations

This paper is concerned with the existence of bounded mild solutions to equations of the form u′(t) = Au(t) + f(t), where A generates a holomorphic semigroup that is not necessarily strongly continuous, and f is a bounded function. This problem arises when one considers a parabolic equation in spaces of continuous functions. The obtained results, that are stated in terms of spectral properties of the spectrum of A and the uniform spectrum of f, extend previous ones.

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about $\mathbb{R}$. Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences. Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.” Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine–Borel Theorem says that the compact sets of $\mathbb{R}^n$ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano–Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.” Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem. Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous. Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli's Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone–Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable. Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

Integral-type operators on continuous function spaces on the real line

Dense Sets in Spaces of Integrable Functions