scholarly journals On rings of Baire one functions

2019 ◽  
Vol 20 (1) ◽  
pp. 237 ◽  
Author(s):  
A. Deb Ray ◽  
Atanu Mondal
Keyword(s):  
Extension Theorem ◽  
Continuous Functions ◽  
Proper Subclass ◽  
Baire One ◽  
Bounded Continuous Functions

<p>This paper introduces the ring of all real valued Baire one functions, denoted by B<sub>1</sub>(X) and also the ring of all real valued bounded Baire one functions, denoted by B<sup>∗</sup><sub>1</sub>(X). Though the resemblance between C(X) and B<sub>1</sub>(X) is the focal theme of this paper, it is observed that unlike C(X) and C<sup>∗</sup>(X) (real valued bounded continuous functions), B<sup>∗</sup><sub>1</sub> (X) is a proper subclass of B<sub>1</sub>(X) in almost every non-trivial situation. Introducing B<sub>1</sub>-embedding and B<sup>∗</sup><sub>1</sub>-embedding, several analogous results, especially, an analogue of Urysohn’s extension theorem is established.</p>

The Black–Scholes Equation with Impulses at Random Times Via Generalized Riemann Integral

2021 ◽  
Vol 15 (01) ◽  
pp. 45-59
Author(s):  
E. M. Bonotto ◽  
M. Federson ◽  
P. Muldowney
Keyword(s):  
Extension Theorem ◽  
Continuous Functions ◽  
Sample Space ◽  
Asset Valuation ◽  
Itô Calculus ◽  
Black Scholes ◽  
Pricing Theory ◽  
Simple Sets ◽  
Random Times ◽  
Riemann Integration

The classical pricing theory requires that the simple sets of outcomes are extended, using the Kolmogorov Extension Theorem, to a sigma-algebra of measurable sets in an infinite-dimensional sample space whose elements are continuous paths; the process involved are represented by appropriate stochastic differential equations (using Itô calculus); a suitable measure for the sample space can be found by means of the Girsanov and Radon–Nikodym Theorems; the derivative asset valuation is determined by means of an expression using Lebesgue integration. It is known that if we replace Lebesgue’s by the generalized Riemann integration to obtain the expectation, the same result can be achieved by elementary methods. In this paper, we consider the Black–Scholes PDE subject to impulse action. We replace the process which follows a geometric Brownian motion by a process which has additional impulsive displacements at random times. Instead of constants, the volatility and the risk-free interest rate are considered as continuous functions which can vary in time. Using the Feynman–Ka[Formula: see text] formulation based on generalized Riemann integration, we obtain a pricing formula for a European call option which copes with many discontinuities. This paper seeks to develop techniques of mathematical analysis in derivative pricing theory which are less constrained by the standard assumption of lognormality of prices. Accordingly, the paper is aimed primarily at analysis rather than finance. An example is given to illustrate the main results.

Strict Topology on Spaces of Continuous Vector-Valued Functions

1979 ◽  
Vol 31 (4) ◽  
pp. 890-896 ◽  
Author(s):  
Seki A. Choo
Keyword(s):  
Tensor Product ◽  
Convex Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Locally Convex Space ◽  
Completely Regular ◽  
Vector Valued Functions ◽  
Bounded Continuous Functions ◽  
Vector Valued ◽  
Locally Convex

In this paper, X denotes a completely regular Hausdorff space, Cb(X) all real-valued bounded continuous functions on X, E a Hausforff locally convex space over reals R, Cb(X, E) all bounded continuous functions from X into E, Cb(X) ⴲ E the tensor product of Cb(X) and E. For locally convex spaces E and F, E ⴲ, F denotes the tensor product with the topology of uniform convergence on sets of the form S X T where S and T are equicontinuous subsets of E′, F′ the topological duals of E, F respectively ([11], p. 96). For a locally convex space G , G ′ will denote its topological dual.

Non-Extendability of Bounded Continuous Functions

1980 ◽  
Vol 32 (4) ◽  
pp. 867-879
Author(s):  
Ronnie Levy
Keyword(s):  
Continuous Function ◽  
Metric Space ◽  
Dense Subset ◽  
Continuous Functions ◽  
Dense Subspace ◽  
Compact Metric Space ◽  
Bounded Continuous Function ◽  
Bounded Functions ◽  
Bounded Continuous Functions ◽  
Completely Metrizable

If X is a dense subspace of Y, much is known about the question of when every bounded continuous real-valued function on X extends to a continuous function on Y. Indeed, this is one of the central topics of [5]. In this paper we are interested in the opposite question: When are there continuous bounded real-valued functions on X which extend to no point of Y – X? (Of course, we cannot hope that every function on X fails to extend since the restrictions to X of continuous functions on Y extend to Y.) In this paper, we show that if Y is a compact metric space and if X is a dense subset of Y, then X admits a bounded continuous function which extends to no point of Y – X if and only if X is completely metrizable. We also show that for certain spaces Y and dense subsets X, the set of bounded functions on X which extend to a point of Y – X form a first category subset of C*(X).

scholarly journals Approximation and extension of continuous functions

1994 ◽  
Vol 57 (2) ◽  
pp. 149-157 ◽  
Author(s):  
Salvador Hernández-Muñoz
Keyword(s):  
Topological Space ◽  
Short Proof ◽  
Extension Theorem ◽  
Continuous Functions ◽  
Vector Valued

AbstractIn this paper we study the approximation of vector valued continuous functions defined on a topological space and we apply this study to different problems. Thus we give a new proof of Machado's Theorem. Also we get a short proof of a Theorem of Katětov and we prove a generalization of Tietze's Extension Theorem for vector-valued continuous functions, thereby solving a question left open by Blair.

scholarly journals Generalized Numerical Index of Function Algebras

2019 ◽  
Vol 2019 ◽  
pp. 1-6
Author(s):  
Han Ju Lee
Keyword(s):  
Banach Space ◽  
Function Algebra ◽  
Complex Banach Space ◽  
Continuous Functions ◽  
Function Algebras ◽  
Bounded Complex ◽  
Closed Subalgebra ◽  
Bounded Continuous Functions ◽  
Complex Valued ◽  
Numerical Index

Let X be a complex Banach space and Cb(Ω:X) be the Banach space of all bounded continuous functions from a Hausdorff space Ω to X, equipped with sup norm. A closed subspace A of Cb(Ω:X) is said to be an X-valued function algebra if it satisfies the following three conditions: (i) A≔{x⁎∘f:f∈A,  x⁎∈X⁎} is a closed subalgebra of Cb(Ω), the Banach space of all bounded complex-valued continuous functions; (ii) ϕ⊗x∈A for all ϕ∈A and x∈X; and (iii) ϕf∈A for every ϕ∈A and for every f∈A. It is shown that k-homogeneous polynomial and analytic numerical index of certain X-valued function algebras are the same as those of X.

scholarly journals Isometries of spaces of unbounded continuous functions

2001 ◽  
Vol 63 (3) ◽  
pp. 475-484
Author(s):  
Jesús Araujo ◽  
Krzysztof Jarosz
Keyword(s):  
Banach Spaces ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Compact Sets ◽  
Surjective Isometry ◽  
Bounded Continuous Functions ◽  
Stone Theorem

By the classical Banach-Stone Theorem any surjective isometry between Banach spaces of bounded continuous functions defined on compact sets is given by a homeomorphism of the domains. We prove that the same description applies to isometries of metric spaces of unbounded continuous functions defined on non compact topological spaces.

scholarly journals Local invertibility in subrings of C*(X)

1992 ◽  
Vol 46 (3) ◽  
pp. 449-458 ◽  
Author(s):  
H. Linda Byun ◽  
Lothar Redlin ◽  
Saleem Watson
Keyword(s):  
Continuous Functions ◽  
Closed Sets ◽  
Complete Ring ◽  
Maximal Ideals ◽  
One To One ◽  
Local Invertibility ◽  
Bounded Continuous Functions ◽  
Ring Of Functions ◽  
Uniformly Closed

It is known that the maximal ideals in the rings C(X) and C*(X) of continuous and bounded continuous functions on X, respectively, are in one-to-one correspondence with βX. We have shown previously that the same is true for any ring A(X) between C(X) and C*(X). Here we consider the problem for rings A(X) contained in C*(X) which are complete rings of functions (that is, they contain the constants, separate points and closed sets, and are uniformly closed). For every noninvertible f ∈ A(X), we define a z–filter ZA(f) on X which, in a sense, provides a measure of where f is ‘locally invertible’. We show that the map ZA generates a correspondence between ideals of A(X) and z–filters on X. Using this correspondence, we construct a unique compactification of X for every complete ring of functions. Each such compactification is explicitly identified as a quotient of βX. In fact, every compactification of X arises from some complete ring of functions A(X) via this construction. We also describe the intersections of the free ideals and of the free maximal ideals in complete rings of functions.

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