Some Results on Maxima and Minima of Real Functions of Vector Variables: A New Perspective

Abstract

In this work, the extreme points of real vector variable functions are obtained without the use of the classical theory that involves the use of partial derivatives. We illustrate with several theorems and examples a new method that consists of establishing an appropriate link between the function to be optimized, its restrictions and the result, stating that: given \(n\) non-zero real numbers \(a_{1},a_{2},\cdots,a_{n}\in\mathbb{R}\), then there exists a unique \(\lambda\in\mathbb{R}\) such that: This relation is obtained by decomposing the Hilbert space \(\mathbb{R}^{n}\) as the direct sum of a closed subspace and its orthogonal complement. Since the dimension of the space \(\mathbb{R}^{n}\) is finite, this guarantees that any linear functional defined on the space \(\mathbb{R}^{n}\) is continuous, and this guarantees that the kernel of said linear functional is closed in the space \(\mathbb{R}^{n}\), therefore we have that the space \(\mathbb{R}^{n}\) breaks down, as the direct sum of the kernel of the continuous linear functional \(f\) and its orthogonal complement, that is: \(\mathbb{R}^{n}\,-\,\ker f\,\bigoplus\,\left[\,\ker f\,\right]^{\perp}\), where the dimension of \(\ker f\,-\,n\,-\,1\) and the dimension of \(\left[\,\ker f\,\right]^{\perp}\,-\,1\). Adding to the link found new definitions about the hierarchy of one variable in relation to another and the fact that if \(x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}\,-\,r^{2}\) then the \(\max\{x_{1}+x_{2}+\cdots+x_{n}\}\,-\,r\sqrt{n}\) and the \(\min\{x_{1}+x_{2}+\cdots+x_{n}\}\,-\,-r\sqrt{n}\) we solve the optimization problem without using classical theory.