On the finite dimensionality of closed subspaces in Lp(M, dμ) ∩ Lq(M, dν)

Balinsky, Alexander A. ORCID: https://orcid.org/0000-0002-8151-4462 and Prykarpatski, Anatolij K. 2021. On the finite dimensionality of closed subspaces in Lp(M, dμ) ∩ Lq(M, dν). Axioms 10 (4) , 275. 10.3390/axioms10040275

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Abstract

Finding effective finite-dimensional criteria for closed subspaces in Lp, endowed with some additional functional constraints, is a well-known and interesting problem. In this work, we are interested in some sufficient constraints on closed functional subspaces, Sp⊂Lp, whose finite dimensionality is not fixed a priori and can not be checked directly. This is often the case in diverse applications, when a closed subspace Sp⊂Lp is constructed by means of some additional conditions and constraints on Lp with no direct exemplification of the functional structure of its elements. We consider a closed topological subspace, S(q)p, of the functional Banach space, Lp(M,dμ), and, moreover, one assumes that additionally, S(q)p⊂Lq(M,dν) is subject to a probability measure ν on M. Then, we show that closed subspaces of Lp(M,dμ)∩Lq(M,dν) for q>max{1,p},p>0 are finite dimensional. The finite dimensionality result concerning the case when q>p>0 is open and needs more sophisticated techniques, mainly based on analysis of the complementary subspaces to Lp(M,dμ)∩Lq(M,dν).

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Additional Information:	This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/)
Publisher:	MDPI
ISSN:	2075-1680
Date of First Compliant Deposit:	2 November 2021
Date of Acceptance:	12 October 2021
Last Modified:	10 May 2023 10:37
URI:	https://orca.cardiff.ac.uk/id/eprint/145259

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