Real-linear isometries between certain subspaces of continuous functions

2013 ◽  
Vol 11 (11) ◽  
Author(s):  
Arya Jamshidi ◽  
Fereshteh Sady
Keyword(s):  
Metric Spaces ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Hausdorff Spaces ◽  
Linear Isometry ◽  
Compact Metric Spaces ◽  
Separating Property ◽  
Similar Description ◽  
Real Linear ◽  
Linear Isometries

AbstractIn this paper we first consider a real-linear isometry T from a certain subspace A of C(X) (endowed with supremum norm) into C(Y) where X and Y are compact Hausdorff spaces and give a result concerning the description of T whenever A is a uniform algebra on X. The result is improved for the case where T(A) is, in addition, a complex subspace of C(Y). We also give a similar description for the case where A is a function space on X and the range of T is a real subspace of C(Y) satisfying a ceratin separating property. Next similar results are obtained for real-linear isometries between spaces of Lipschitz functions on compact metric spaces endowed with a certain complete norm.

scholarly journals A new proof of the Banach-stone theorem

1998 ◽  
Vol 57 (1) ◽  
pp. 55-58
Author(s):  
N.J. Cutland ◽  
G.B. Zimmer
Keyword(s):  
Metric Spaces ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Original Proof ◽  
Compact Metric Spaces ◽  
Peak Point ◽  
Standard Part ◽  
Peak Functions ◽  
Stone Theorem

Banach's original proof of the Banach-Stone theorem for compact metric spaces uses peak functions, that is, continuous functions which assume their norm in just one point. We show by using nonstandard methods that the peak point approach also works for compact Hausdorff spaces. The peak functions are replaced by internal functions whose standard part is supported in one monad.

scholarly journals Real-linear isometries between subspaces of continuous functions

2014 ◽  
Vol 413 (1) ◽  
pp. 229-241 ◽  
Author(s):  
Hironao Koshimizu ◽  
Takeshi Miura ◽  
Hiroyuki Takagi ◽  
Sin-Ei Takahasi
Keyword(s):  
Continuous Functions ◽  
Real Linear ◽  
Linear Isometries

scholarly journals A note on finite codimensional linear isometries ofC(X)intoC(Y)

1995 ◽  
Vol 18 (4) ◽  
pp. 677-680 ◽  
Author(s):  
Sin-Ei Takahasi ◽  
Takateru Okayasu
Keyword(s):  
Finite Codimension ◽  
Hausdorff Spaces ◽  
Linear Isometry ◽  
Continuous Maps ◽  
Linear Isometries

Let(X,Y)be a pair of compact Hausdorff spaces. It is shown that a certain property of the class of continuous maps ofYontoXis equivalent to the non-existence of linear isometry ofC(X)intoC(Y)whose range has finite codimension>0.

scholarly journals Isomorphisms between spaces of vector-valued continuous functions

1983 ◽  
Vol 26 (1) ◽  
pp. 29-48 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Banach Space ◽  
Metric Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Compact Metric Spaces ◽  
Vector Valued

A theorem due to Milutin [12] (see also [13]) asserts that for any two uncountable compact metric spaces Ω1 and Ω2 the spaces of continuous real-valued functions C(Ω1) and C(Ω2) are linearly isomorphic. It immediately follows from consideration of tensor products that if X is any Banach space then C(Ω1;X) and C(Ω2;X) are isomorphic.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

scholarly journals The Blum-Hanson Property

2019 ◽  
Vol 6 (1) ◽  
pp. 92-105
Author(s):  
Sophie Grivaux
Keyword(s):  
Banach Space ◽  
Hilbert Spaces ◽  
Separable Banach Space ◽  
Metric Spaces ◽  
Compact Metric Spaces ◽  
Theoretic Motivation

AbstractGiven a (real or complex, separable) Banach space, and a contraction T on X, we say that T has the Blum-Hanson property if whenever x, y ∈ X are such that Tnx tends weakly to y in X as n tends to infinity, the means{1 \over N}\sum\limits_{k = 1}^N {{T^{{n_k}}}x} tend to y in norm for every strictly increasing sequence (nk) k≥1 of integers. The space X itself has the Blum-Hanson property if every contraction on X has the Blum-Hanson property. We explain the ergodic-theoretic motivation for the Blum-Hanson property, prove that Hilbert spaces have the Blum-Hanson property, and then present a recent criterion of a geometric flavor, due to Lefèvre-Matheron-Primot, which allows to retrieve essentially all the known examples of spaces with the Blum-Hanson property. Lastly, following Lefèvre-Matheron, we characterize the compact metric spaces K such that the space C(K) has the Blum-Hanson property.

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