On the Finite Dimensionality of Closed Subspaces in Lp(M, dμ) ∩ Lq(M, dν)

Alexander A Balinsky,Anatolij K Prykarpatski

doi:10.3390/axioms10040275

Alexander A Balinsky, Anatolij K Prykarpatski

Open Access

https://doi.org/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, Sp(q), of the functional Banach space, Lp(M,dμ), and, moreover, one assumes that additionally, Sp(q)⊂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).

Highlights

The problems, concerned with the finite dimensionality of closed functional subspaces in L p (in part, in L p (0, 1; C)), are of long-time interest in analysis, being related to their many applications in operator and approximation theories [1,2,3,4,5], in dynamical systems theory [6,7,8,9,10,11] and other applied fields
We show that closed subspaces of L p ( M, dμ) ∩ Lq ( M, dν) for q > max{1, p}, p > 0 are finite dimensional
One can recall a central problem in Banach space theory to classify the complemented subspaces of L p up to isomorphism; the finite-dimensional analogue is to find for any given S p ⊂ L p a description of the finitedimensional spaces which are S p -isomorphic to S p -complemented subspaces of L p

Summary

Introduction

The problems, concerned with the finite dimensionality of closed functional subspaces in L p (in part, in L p (0, 1; C)), are of long-time interest in analysis, being related to their many applications in operator and approximation theories [1,2,3,4,5], in dynamical systems theory [6,7,8,9,10,11] and other applied fields. Dimensionality of Closed Subspaces in L p ( M, dμ) ∩ Lq ( M, dν). We show that closed subspaces of L p ( M, dμ) ∩ Lq ( M, dν) for q > max{1, p}, p > 0 are finite dimensional.

Results

Conclusion