On the Infimum of Certain Functionals

Abstract

Here is a particular case of our main result: Let X be a real Banach space, $\varphi: X \rightarrow \mathbf{R}$ a nonzero continuous linear functional and ψ: X → R a nonconstant Lipschitzian functional with Lipschitz constant equal to $\|\varphi \|_{X^{{\ast}}}$. Then, we have

$$\displaystyle\begin{array}{rcl} & & \max \left \{\inf _{x\in X}(\varphi (x) +\psi (x)),\inf _{x\in X}(\varphi (x) -\psi (x))\right \} {}\\ & & \quad =\inf _{x\in X}(\varphi (x) + \vert \psi (x)\vert ) =\liminf _{\|x\|\rightarrow +\infty }(\varphi (x) + \vert \psi (x)\vert ) {}\\ \end{array}$$