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
Received December 12, 2024
Revised January 15, 2025
Accepted January 20, 2025
Vadim Remirovich Lazarev, Cand. Sci. (Phys.-Math.), Faculty of Mechanics and Mathematics, Tomsk State University, Tomsk, 634050 Russia, e-mail: lazarev@math.tsu.ru
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Cite this article as: V.R. Lazarev. Comparison of spaces of functionals with finite support. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 1, pp. 101–109.
