On the infimum of certain functionals

TL;DR

This paper proves a new equality involving infima of certain functionals on Banach spaces, specifically relating to Lipschitz functionals and their absolute values, with implications for the behavior at infinity.

Contribution

It establishes a novel equality for infima of combined functionals involving Lipschitz functions and linear functionals in Banach spaces.

Findings

The equality between the maximum of two infima and the infimum of a sum involving absolute values.

The limit inferior of the sum functional equals its infimum at infinity.

Results apply to Lipschitz functionals with specific Lipschitz constants.

Abstract

In this note, in particular, we establish the following result: Let $X$ be a real Banach space, $\varphi\in X^*\setminus \{0\}$ and $\psi:X\to {\bf R}$ a Lipschitzian functional with Lipschitz constant equal to $\varphi\|_X^{*}$. Then, we have $$\max\left\{\inf_{x\in X}(\varphi(x)+\psi(x)),\inf_{x\in X}(\varphi(x)-\psi(x))\right\}=\inf_{x\in X}(\varphi(x)+|\psi(x)|)$$ and $$\liminf_{\|x\|\to +\infty}(\varphi(x)+|\psi(x)|)=\inf_{x\in X}(\varphi(x)+|\psi(x)|)\ .$$