scholarly journals A Remark on the Continuity of the Dual Process

1968 ◽  
Vol 32 ◽  
pp. 287-295 ◽  
Author(s):  
Mamoru Kanda
Keyword(s):  
Banach Space ◽  
Compact Support ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Finite Measure ◽  
Uniform Norm ◽  
Continuous Functions ◽  
Dual Process ◽  
Space Of Continuous Functions ◽  
Locally Compact

Let S be a locally compact (not compact) Hausdorff space satisfying the second axiom of countability and let ℬ be the σ field of all Borel subsets of S and let A be the σ-field of all the subsets of S which, for each finite measure μ defined on (S, A), are in the completed σ field of ℬ relative to μ. We denote by C0 the Banach space of continuous functions vanishing at infinity with the uniform norm and Bk the space of bounded A-measurable functions with compact support in S.

scholarly journals Isomorphisms from Extremely Regular Subspaces of C0K into C0S,X Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Manuel Felipe Cerpa-Torres ◽  
Michael A. Rincón-Villamizar
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Geometrical Parameter ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Continuous Image ◽  
Topological Properties ◽  
Locally Compact ◽  
Regular Subspace ◽  
Locally Compact Hausdorff Space

For a locally compact Hausdorff space K and a Banach space X, let C0K,X be the Banach space of all X-valued continuous functions defined on K, which vanish at infinite provided with the sup norm. If X is ℝ, we denote C0K,X as C0K. If AK be an extremely regular subspace of C0K and T:AK⟶C0S,X is an into isomorphism, what can be said about the set-theoretical or topological properties of K and S? Answering the question, we will prove that if X contains no copy of c0, then the cardinality of K is less than that of S. Moreover, if TT−1<3 and AK is also a subalgebra of C0K, the cardinality of the αth derivative of K is less than that of the αth derivative of S, for each ordinal α. Finally, if λX>1 and TT−1<λX, then K is a continuous image of a subspace of S. Here, λX is the geometrical parameter introduced by Jarosz in 1989: λX=infmaxx+λy:λ=1:x=y=1. As a consequence, we improve classical results about into isomorphisms from extremely regular subspaces already obtained by Cengiz.

A Banach–Stone theorem for spaces of weak* continuous functions

1985 ◽  
Vol 101 (3-4) ◽  
pp. 203-206 ◽  
Author(s):  
Michael Cambern
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Hausdorff Spaces ◽  
Space Of Continuous Functions ◽  
Dual Banach Space ◽  
Stone Theorem

SynopsisIf X is a compact Hausdorff space and E a dual Banach space, let C(X, Eσ*) denote the Banach space of continuous functions F from X to E when the latter space is provided with its weak * topology, normed by . It is shown that if X and Y are extremally disconnected compact Hausdorff spaces and E is a uniformly convex Banach space, then the existence of an isometry between C(X, Eσ*) and C(Y, Eσ*) implies that X and Y are homeomorphic.

On Additive Operators

1971 ◽  
Vol 23 (3) ◽  
pp. 468-480 ◽  
Author(s):  
N. A. Friedman ◽  
A. E. Tong
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Weak Topology ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Additive Functional ◽  
Representation Theorems ◽  
Image Position ◽  
Vector Valued ◽  
Kernel Representation

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):

scholarly journals STRONGLY BOUNDED REPRESENTING MEASURES AND CONVERGENCE THEOREMS

2010 ◽  
Vol 52 (3) ◽  
pp. 435-445 ◽  
Author(s):  
IOANA GHENCIU ◽  
PAUL LEWIS
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Convergence Theorems

AbstractLet K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: K → X endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).

scholarly journals Factoring absolutely summing operators through Hilbert-Schmidt operators

1989 ◽  
Vol 31 (2) ◽  
pp. 131-135 ◽  
Author(s):  
Hans Jarchow
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hausdorff Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Reflexive Banach Spaces ◽  
Summing Operator ◽  
Schmidt Operator ◽  
Second Dual

Let K be a compact Hausdorff space, and let C(K) be the corresponding Banach space of continuous functions on K. It is well-known that every 1-summing operator S:C(K)→l2 is also nuclear, and therefore factors S = S1S2, with S1:l2→l2 a Hilbert–Schmidt operator and S1:C(K)→l2 a bounded operator. It is easily seen that this latter property is preserved when C(K) is replaced by any quotient, and that a Banach space X enjoys this property if and only if its second dual, X**, does. This led A. Pełczyński [15] to ask if the second dual of a Banach space X must be isomorphic to a quotient of a C(K)-space if X has the property that every 1-summing operator X-→l2 factors through a Hilbert-Schmidt operator. In this paper, we shall first of all reformulate the question in an appropriate manner and then show that counter-examples are available among super-reflexive Tsirelson-like spaces as well as among quasi-reflexive Banach spaces.

scholarly journals Distinguishedness of weighted Fréchet spaces of continuous functions

1992 ◽  
Vol 35 (2) ◽  
pp. 271-283 ◽  
Author(s):  
Françoise Bastin
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Density Condition ◽  
Locally Compact ◽  
Spaces Of Continuous Functions ◽  
Locally Compact Hausdorff Space

In this paper, we prove that if is an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff space X such that then the Fréchet space C(X) is distinguished if and only if it satisfies Heinrich's density condition, or equivalently, if and only if the sequence satisfies condition (H) (cf. e.g.‵[1] for the introduction of (H)). As a consequence, the bidual λ∞(A) of the distinguished Köthe echelon space λ0(A) is distinguished if and only if the space λ1(A) is distinguished. This gives counterexamples to a problem of Grothendieck in the context of Köthe echelon spaces.

scholarly journals Double-dual types over the Banach spaceC(K)

2005 ◽  
Vol 2005 (16) ◽  
pp. 2533-2545
Author(s):  
Markus Pomper
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Lower Semicontinuous ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Isolated Points

LetKbe a compact Hausdorff space andC(K)the Banach space of all real-valued continuous functions onK, with the sup-norm. Types overC(K)(in the sense of Krivine and Maurey) can be uniquely represented by pairs(ℓ,u)of bounded real-valued functions onK, whereℓis lower semicontinuous,uis upper semicontinuous,ℓ≤u, andℓ(x)=u(x)for all isolated pointsxofK. A condition that characterizes the pairs(ℓ,u)that represent double-dual types overC(K)is given.

Types over c(k) spaces

2004 ◽  
Vol 77 (1) ◽  
pp. 17-28
Author(s):  
Markus Pomper
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Lower Semicontinuous ◽  
Continuous Functions ◽  
Upper Semicontinuous ◽  
Bounded Functions ◽  
Isolated Point

AbstractLet K be a compact Hausdorff space and C(K) the Banach space of all real-valued continuous functions on K, with the sup norm. Types over C(K) (in the sense of Krivine and Maurey) are represented here by pairs (l, u) of bounded real-valued functions on K, where l is lower semicontinuous and u is upper semicontinuous, l ≤ u and l(x) = u(x) for every isolated point x of K. For each pair the corresponding type is defined by the equation τ(g) = max{║l + g║∞, ║u + g║∞} for all g ∈ C(K), where ║·║∞ is the sup norm on bounded functions. The correspondence between types and pairs (l, u) is bijective.

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