Comparison of spaces of functionals with finite support

V. R. Lazarev

Tomsk State University

Abstract: In this paper, a functional is understood as any continuous real-valued function $f$ on $C_p(X)$ such that $f(0)=0$. The space $FS(X)$ of functionals with finite support and its subspace $\hat{L}_p(X)$ are studied. These spaces are compared with the space of linear continuous functionals $L_p(X)$. A theorem on the general form of a functional with finite support is proved. The theorem is used to show that the three mentioned spaces are pairwise distinct. It is also proved that $FS(X)$ is everywhere dense in the space of all functionals and that $\hat{L}_p(X)$ is nowhere dense in the space of all functionals, but the sum $L_p(X)+ \hat{L}_p(X)$ is dense in that space. The latter fact implies that the space $\hat{L}_p(X)$ is essentially wider than $L_p(X)$. The functional space $\hat{L}_p(X)$ defines some class $\hat LH$ of homeomorphisms of spaces of continuous functions, similarly to how the space $L_p(X)$ defines the class of linear homeomorphisms. It is already known that homeomorphisms of the class $\hat LH$ preserve the Lindelöf number of domains. We prove that a homeomorphism of the class $\hat LH$ cannot always be replaced by a linear one. Hence we have a generalization of Bouziad's known theorem on the $l$-invariance of the Lindelöf number.

Keywords: pointwise convergence topology, functional with finite support, Lindelöf number, $l$-equivalence.

UDC: 515.12

MSC: 54C35

Received: 12.12.2024
Revised: 15.01.2025
Accepted: 20.01.2025

DOI: 10.21538/0134-4889-2025-31-1-101-109

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