Norm On The Variable Mixed Space $ \ell^{q(\cdot)}(L^{p(\cdot)}) $

TL;DR

This paper investigates the fundamental norm properties of the variable mixed space \( \ell^{q(\cdot)}(L^{p(\cdot)}) \), addressing an open problem in the mathematical analysis of variable exponent spaces.

Contribution

It establishes the norm properties of the variable mixed space \( \ell^{q(\cdot)}(L^{p(\cdot)}) \), solving an open problem posed by Almeida and H"asto.

Findings

Confirmed the norm properties of \( \ell^{q(\cdot)}(L^{p(\cdot)}) \) spaces.

Extended the understanding of variable exponent function spaces.

Addressed and solved an open problem in the field.

Abstract

This paper aims to establish the norm properties of the variable mixed space $ \ell^{q(\cdot)}(L^{p(\cdot)}) $ when $ 1<q_-,p_-,q_+,p_+<\infty $. In this way, we address the open problem raised by Almeida and H\"{a}st\"{o}.