# On linear continuous operators between distinguished spaces $$C_p(X)$$

@article{Kakol2021OnLC, title={On linear continuous operators between distinguished spaces \$\$C\_p(X)\$\$}, author={Jerzy Kakol and Arkady Leiderman}, journal={Revista de la Real Academia de Ciencias Exactas, F{\'i}sicas y Naturales. Serie A. Matem{\'a}ticas}, year={2021} }

As proved in [16], for a Tychonoff space X , a locally convex space Cp(X) is distinguished if and only if X is a ∆-space. If there exists a linear continuous surjective mapping T : Cp(X) → Cp(Y ) and Cp(X) is distinguished, then Cp(Y ) also is distinguished [17]. Firstly, in this paper we explore the following question: Under which conditions the operator T : Cp(X) → Cp(Y ) above is open? Secondly, we devote a special attention to concrete distinguished spaces Cp([1, α]), where α is a countable…

## One Citation

A note on Banach spaces $E$ admitting a continuous map from $C_p(X)$ onto $E_{w}$

- Mathematics
- 2021

Cp(X) denotes the space of continuous real-valued functions on a Tychonoff space X endowed with the topology of pointwise convergence. A Banach space E equipped with the weak topology is denoted by…

## References

SHOWING 1-10 OF 43 REFERENCES

A characterization of $X$ for which spaces $C_p(X)$ are distinguished and its applications

- Mathematics
- 2020

We prove that the locally convex space $C_{p}(X)$ of continuous real-valued functions on a Tychonoff space $X$ equipped with the topology of pointwise convergence is distinguished if and only if $X$…

On the weak and pointwise topologies in function spaces

- Mathematics
- 2015

For a compact space K we denote by $$C_w(K)$$Cw(K) ($$C_p(K)$$Cp(K)) the space of continuous real-valued functions on K endowed with the weak (pointwise) topology. In this paper we address the…

Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

- MathematicsRevista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
- 2019

The famous Rosenthal–Lacey theorem asserts that for each infinite compact space K the Banach space C(K) admits a quotient isomorphic to Banach spaces c or $$\ell _{2}$$ℓ2. The aim of the paper is to…

A metrizable X with Cp(X) not homeomorphic to Cp(X) × Cp(X)

- Mathematics
- 2016

We give an example of an infinite metrizable space X such that the space Cp(X), of continuous real-valued functions on X endowed with the pointwise topology, is not homeomorphic to its own square…

On linear continuous open surjections of the spaces CP(X)

- Mathematics
- 1997

Solving a problem by Arkhangel'skiĭ, we construct a linear continuous open surjection L : Cp(X) → CP(Y) for compacta X and Y with 0 < dim X < dim Y < ∞. An example of nonopen linear continuous…

Distinguished $$ C_{p}(X) $$ spaces

- Mathematics
- 2020

We continue our initial study of
$$C_{p}(X) $$
spaces that are distinguished, equiv., are large subspaces of
$$\mathbb {R}^{X}$$
, equiv., whose strong duals
$$L_{\beta }( X) $$
carry the…

C(K) quotients of separable ℒ∞ spaces

- Mathematics
- 1979

It is shown that ifX is an ℒ⫗ space with separable dual, thenX has a quotient isomorphic toC(ω)α if, and only if, there is anɛ >0, such that theɛ-Szlenk index ofX is at leastα. It was previously…