Space Of Real-valued Continuous Functions Research Articles

The sequential topology on is not regular

The compact-open topology on the set of continuous functionals from the Baire space to the natural numbers is well known to be zero-dimensional. We prove that the closely related sequential topology on this set is not even regular. The sequential topology arises naturally as the topology carried by the exponential formed in various cartesian closed categories of topological spaces. Moreover, we give an example of an effectively open subset of that violates regularity. The topological properties of are known to be closely related to an open problem in Computable Analysis. We also show that the sequential topology on the space of continuous real-valued functions on a Polish space need not be regular.

The Basic Theorem and its Consequences

Let T be a compact Hausdor topological space and let M denote an n-dimensional subspace of the space C(T ), the space of real-valued continuous functions on T and let the space be equipped with the uniform norm. Zukhovitskii (7) attributes the Basic Theorem to E.Ya.Remez and gives a proof by duality. He also gives a proof due to Shnirel'man, which uses Helly's Theorem, now the paper obtains a new proof of the Basic Theorem. The significance of the Basic Theorem for us is that it reduces the characterization of a best approximation to f 2 C(T ) from M to the case of finite T , that is to the case of approximation in l 1 (r). If one solves the problem for the finite case of T then one can deduce the solution to the general case. An immediate consequence of the Basic Theorem is that for a finite dimensional subspace M of C0(T ) there exists a separating measure for M and f 2 C0(T )\M, the cardinality of whose support is not greater than dimM+1. This result is a special case of a more general abstract result due to Singer (5). Then the Basic Theorem is used to obtain a general characterization theorem of a best approximation from M to f 2 C(T ). We also use the Basic Theorem to establish the suciency of Haar's condition for a subspace M of C(T ) to be Chebyshev.

A NOTE ON SPACES Cp(X)K-ANALYTIC-FRAMED IN ℝX

AbstractThis paper characterizes the K-analyticity-framedness in ℝX for Cp(X) (the space of real-valued continuous functions on X with pointwise topology) in terms of Cp(X). This is used to extend Tkachuk’s result about the K-analyticity of spaces Cp(X) and to supplement the Arkhangel ′skiĭ–Calbrix characterization of σ-compact cosmic spaces. A partial answer to an Arkhangel ′skiĭ–Calbrix problem is also provided.

The quantitative difference between countable compactness and compactness

We establish here some inequalities between distances of pointwise bounded subsets H of R X to the space of real-valued continuous functions C ( X ) that allow us to examine the quantitative difference between (pointwise) countable compactness and compactness of H relative to C ( X ) . We prove, amongst other things, that if X is a countably K-determined space the worst distance of the pointwise closure H ¯ of H to C ( X ) is at most 5 times the worst distance of the sets of cluster points of sequences in H to C ( X ) : here distance refers to the metric of uniform convergence in R X . We study the quantitative behavior of sequences in H approximating points in H ¯ . As a particular case we obtain the results known about angelicity for these C p ( X ) spaces obtained by Orihuela. We indeed prove our results for spaces C ( X , Z ) (hence for Banach-valued functions) and we give examples that show when our estimates are sharp.

Domain representability of Cp(X)

Let $C_{\rm p}(X)$ be the space of continuous real-valued functions on $X$, with the topology of pointwise convergence. We consider the following three properties of a space $X$: (a) $C_{\rm p}(X)$ is Scott-domain representable; (b) $C_{\rm p}(X)$

A characterization of trans-separable spaces

The paper shows that a uniform space $X$ is trans-separable if and only if every pointwise bounded uniformly equicontinuous subset of the space of continuous real-valued functions $C_{c}(X) $ equipped with the compact-open topology is metrizable. This extends earlier results of Pfister and Robertson and also applies to show that if $C_{c}(X) $ is angelic then $X$ is trans-separable. The precise relation among DCCC spaces and trans-separable spaces has been also determined.

Some properties of set-open topologies

The space of continuous real-valued functions with set-open topology is studied. The coincidence of the set-open topologies for various families λ is considered and conditions for a set-open topology to coincide with the uniform topology are investigated. The separation axioms in the spaces C λ(X) are studied. Criteria for the local compactness and σ-compactness of C λ(X) are given.

Distance to spaces of continuous functions

We link here distances between iterated limits, oscillations, and distances to spaces of continuous functions. For a compact space K, a uniformly bounded set H of the space of real-valued continuous functions C ( K ) , and ε ⩾ 0 , we say that H ε-interchanges limits with K, if the inequality | lim n lim m f m ( x n ) − lim m lim n f m ( x n ) | ⩽ ε holds for any two sequences ( x n ) in K and ( f m ) in H, provided the iterated limits exist. We prove that H ε-interchanges limits with K if and only if the inequality for the oscillations osc ∗ ( f ) = sup x ∈ K osc ∗ ( f , x ) = sup x ∈ K inf { sup y ∈ U | f ( y ) − f ( x ) | : U neighb. of x } ⩽ ε , holds for every f in the closure cl R K ( H ) of H in R K . Since oscillations actually measure distances to spaces of continuous functions, we get that if H ε-interchanges limits with K, then d ˆ ( cl R K ( H ) , C ( K ) ) : = sup f ∈ cl R K ( H ) d ( f , C ( K ) ) ⩽ ε . Conversely, if d ˆ ( cl R K ( H ) , C ( K ) ) ⩽ ε , then H 2 ε-interchanges limits with K. We also prove that H ε-interchanges limits with K if and only if its convex hull conv ( H ) does. As a consequence we obtain that for each uniformly bounded pointwise compact subset H of R K we have d ˆ ( cl R K ( conv ( H ) ) , C ( K ) ) ⩽ 5 d ˆ ( H , C ( K ) ) . The above estimates can be applied to measure distances from elements of the bidual E * * to the Banach space E: for a w * -compact subset H of E * * , we have d ˆ ( w * - cl ( conv ( H ) ) , E ) ⩽ 5 d ˆ ( H , E ) . These results are quantitative versions of the classical Eberlein–Grothendieck and Krein–Smulyan theorems. In the case of Banach spaces these quantitative generalizations have been recently studied by M. Fabian, A.S. Granero, P. Hajék, V. Montesionos, and V. Zizler. Our topological approach allows us to go further: most of the above statements remain true for spaces C ( X , Z ) for a paracompact (in some cases, a normal countably compact) space X and a convex compact subset Z of a Banach space.

Spaces of continuous functions over a Ψ-space

The Lindelöf property of the space of continuous real-valued continuous functions is studied. A consistent example of an uncountable Ψ-like space is constructed for which the space of continuous real-valued functions with the pointwise convergence topology is Lindelöf.

Geoffrion proper efficiency in an infinite dimensional space

We present a generalization of Geoffrions proper efficiency in the space of continuous real-valued functions on a compact set T. As an important advantage, our definition keeps the componentwise structure used in the original definition. Further, we investigate the relations between our generalized Geoffrion proper efficiency and other types of proper efficiency such as those of Borwein and Benson or points received via linear scalarization.

Function-space compactifications of function spaces

If X and Y are Hausdorff spaces with X locally compact, then the compact-open topology on the set C(X,Y) of continuous maps from X to Y is known to produce the right function-space topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for any Tychonoff space Y, there is a densely injective space Z containing Y as a densely embedded subspace such that, for every locally compact space X, the set C(X,Z) has a compact Hausdorff topology whose relative topology on C(X,Y) is the compact-open topology. The following are derived as corollaries: (1) If X and Y are compact Hausdorff spaces then C(X,Y) under the compact-open topology is embedded into the Vietoris hyperspace V(X×Y). (2) The space of real-valued continuous functions on a locally compact Hausdorff space under the compact-open topology is embedded into a compact Hausdorff space whose points are pairs of extended real-valued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways.

Approximations from Subspaces of C0(X)

We show among other things that if B is a linear space of continuous real-valued functions vanishing at infinity on a locally compact Hausdorff space X, for which there is a continuous function h defined in a neighbourhood of 0 in the real line which is non-affine in every neighbourhood of 0 and satisfies |h(t)|⩽k|t| for all t, such that hb is in B whenever b is in B and the composite function is defined, then every function in C0(X) which can be approximated on every pair of points in X by functions in B can be approximated uniformly by functions in B.

Le degré de Lindelöf est $l$-invariant

The Lindelof number $l(X)$ of a Tychonoff space $X$ is the smallest infinite cardinal $\tau$ such that any open cover of $X$ contains a subcover of cardinality less than or equal to $\tau$. The symbol $C_p(X)$ denotes the space of real-valued continuous functions on $X$ endowed with the topology of simple convergence. A well known fact is that if $C_p(X)$ and $C_p(Y)$ are isomorphic as topological rings, then $X$ and $Y$ are homeomorphic. The main resul of this paper shows that if $C_p(X)$ and $C_p(Y)$ are linearly homeomorphic, then $l(X)=l(Y)$.

On a Problem of G. G. Lorentz

Let C(B) denote the space of real-valued continuous functions on B. At the conference “Harmonic Analysis and Approximations” held at Nor Amberd in Armenia in September, 1998, the following general problem was posed by Professor George G. Lorentz: Find conditions on finite-dimensional subspaces U and V so that U/V is a uniqueness set in the problem of best uniform approximation to elements of C(B). In this paper we consider this problem.

Existence of periodic traveling wave solution to the forced generalized nearly concentricKorteweg‐de Vries equation

This paper is concerned with periodic traveling wave solutions of the forced generalized nearly concentric Korteweg‐de Vries equation in the form of . The authors first convert this equation into a forced generalized Kadomtsev‐Petviashvili equation, , and then to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relationship between the ordinary differential equation and nonlinear integral equations with symmetric kernels is established by using the Green′s function method. The integral representations generate compact operators in a Banach space of real‐valued continuous functions. The Schauder′s fixed point theorem is then used to prove the existence of nonconstant solutions to the integral equations. Therefore, the existence of periodic traveling wave solutions to the forced generalized KP equation, and hence the nearly concentric KdV equation, is proved.

Near metric properties of function spaces

“Near metric” properties of the space of continuous real-valued functions on a space X with the compact-open topology or with the topology of pointwise convergence are examined. In particular, it is investigated when these spaces are stratifiable or cometrisable.

A Hilbert cube compactification of the Banach space of continuous functions

Let C( X) be the Banach space of continuous real-valued functions of an infinite compactum X with the sup-norm, which is homeomorphic to the pseudo-interior s = (−1, 1) ω of the Hilbert cube Q = [−1, 1] ω . We can regard C( X) as a subspace of the hyperspace exp(X × R ̄ ) of nonempty compact subsets of X × R ̄ endowed with the Vietoris topology, where R ̄ = [−∞, ∞] is the extended real line (cf. (Fedorchuk, 1991)). Then the closure R ̄ (X) of C( X) in exp(X × R ̄ ) is a compactification of C( X). We show that the pair ( C ̄ (X), C(X)) is homeomorphic to ( Q, s) if X is locally connected. As a corollary, we give the affirmative answer to a question of Fedorchuk (Fedorchuk, 1996, Question 2.6).

Functional Calculus for Banach Function Algebras and Banach Function Spaces of Continuous Functions Vanishing at Infinity

We show among other things that ifBis a Banach function space of continuous real-valued functions vanishing at infinity on a locally compact Hausdorff spaceX, with the property that for some odd natural numberp>1,b1/p∈Bfor allb∈B, thenB=C0(X).

A Korovkin type theorem for weighted spaces of continuous functions

We prove a Korovkin type approximation theorem for positive linear operators on weighted spaces of continuous real-valued functions on a compact Hausdorff space X. These spaces comprise a variety of subspaces of C (X) with suitable locally convex topologies and were introduced by Nachbin 1967 and Prolla 1977. Some early Korovkin type results on the weighted approximation of real-valued functions in one and several variables with a single weight function are due to Gadzhiev 1976 and 1980.

Baire trees, bad norms and the Namioka property

Many positive results are known to hold for the class of Banach spaces known as Asplund spaces and it was for a time conjectured that Asplund spaces should admit equivalent norms with good smoothness and strict convexity properties. A counterexample to these conjectures, in the form of a space of continuous real-valued functions on a suitably chosen tree, was presented in [5]. In this paper we show that the bad behaviour of that example is shared by a wider class of Banach spaces, associated with a wider class of trees. The immediate aim of this extension of the original result is to answer a question posed by Deville and Godefroy [3]. They introduced and studied a subclass of Asplund spaces, those with Corson compact bidual balls, and asked whether this additional assumption is enough to guarantee the existence of nice renormings. We show that it is not.