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On a Lemma by Brézis and Haraux

Minh N. Bùi

Abstract

We propose several applications of an often overlooked part of the 1976 paper by Brézis and Haraux, in which the Brézis--Haraux theorem was established. Our results unify and extend various existing ones on the range of a linearly composite monotone operator and provide new insight into their seminal paper.

On a Lemma by Brézis and Haraux

Abstract

We propose several applications of an often overlooked part of the 1976 paper by Brézis and Haraux, in which the Brézis--Haraux theorem was established. Our results unify and extend various existing ones on the range of a linearly composite monotone operator and provide new insight into their seminal paper.
Paper Structure (6 theorems, 42 equations)

This paper contains 6 theorems, 42 equations.

Key Result

Lemma 2

Let $C$ and $D$ be nonempty subsets of $\mathcal{H}$ such that $\mathop{\mathrm{conv}}\nolimits D\subset\overline{C}$ and $\mathop{\mathrm{rint}}\nolimits(\mathop{\mathrm{conv}}\nolimits D)\subset C\subset D$. Then $C\simeq D\simeq\mathop{\mathrm{conv}}\nolimits D$.

Theorems & Definitions (22)

  • Lemma 2
  • proof
  • Remark 3
  • Lemma 4: Brézis--Haraux
  • Corollary 5
  • Remark 6
  • Example 7
  • proof
  • Theorem 8
  • proof
  • ...and 12 more