In this paper, we first give two versions of the Fenchel duality formula and Brøndsted-Rockafellar's theorem. The versions are useful and the proofs are not so difficult. Next, we present a theorem in a Banach space to reconsider the Ekeland variational principle and Takahashi's nonconvex minimization theorem proved in a complete metric space. Finally, using the version of the Fenchel duality formula, the Ekeland variational principle and the version of Brøndsted-Rockafellar's theorem, we prove a minimization theorem in a Banach space and then present a short proof of well–known Rockafellar's theorem for maximal monotonicity of subdifferentials.

在本文中，我们首先给出了 Fenchel 对偶公式和 Brøndsted-Rockafellar 定理的两个版本。这些版本很有用，证明也不难。接下来，我们在 Banach 空间中提出一个定理，以重新考虑在完全度量空间中证明的 Ekeland 变分原理和 Takahashi 的非凸最小化定理。最后，使用 Fenchel 对偶公式的版本、Ekeland 变分原理和 Brøndsted-Rockafellar 定理的版本，我们证明了 Banach 空间中的最小化定理，然后给出了著名的 Rockafellar 定理的简短证明，用于子微分的最大单调性.