# A Functional Analytic Perspective to the div-curl Lemma

## TL;DR

This paper offers a new abstract functional analytic framework for the div-curl lemma, connecting it to operator sequences in Hilbert spaces and differential forms, with implications for biharmonic operators.

## Contribution

It introduces a novel functional analytic formulation of the div-curl lemma, linking it to operator sequences and differential forms in Hilbert spaces.

## Key findings

- Provides an abstract operator-theoretic formulation of the div-curl lemma.
- Connects the div-curl lemma to differential forms and recent biharmonic operator sequences.
- Enhances understanding of the lemma's structure through functional analysis.

## Abstract

We present an abstract functional analytic formulation of the celebrated $\dive$-$\curl$ lemma found by F.~Murat and L.~Tartar. The viewpoint in this note relies on sequences for operators in Hilbert spaces. Hence, we draw the functional analytic relation of the div-curl lemma to differential forms and other sequences such as the $\Grad\grad$-sequence discovered recently by D.~Pauly and W.~Zulehner in connection with the biharmonic operator.