Embedding between Lebesgue and weak Lebesgue sequence spaces is strictly singular

TL;DR

This paper proves that the natural embedding from Lebesgue sequence spaces to weak Lebesgue sequence spaces is strictly singular, extending classical results about embeddings between Lebesgue spaces.

Contribution

It extends the classical result of strict singularity from Lebesgue space embeddings to include the non-compact embedding into weak Lebesgue spaces.

Findings

The embedding $ell_{p} o ell_{p, hinspace ext{infty}}$ is strictly singular.

This extends the understanding of embeddings between Lebesgue and weak Lebesgue spaces.

The result generalizes classical theorems about Lebesgue space embeddings.

Abstract

Given $1< p < q < \infty$ it is well know that the natural embedding of Lebesgue sequence spaces $\ell_p \hookrightarrow \ell_q$ is strictly singular. In this paper we extend this classical results and show that even the natural non-compact embedding between Lebesgue and weak Lebesgue sequence spaces $\ell_{p}\hookrightarrow \ell_{p,\infty}$ is strictly singular.