# The weak n-inner product space

## TL;DR

This paper introduces the concept of weak n-inner products, generalizes the n-iterated 2-inner product, and explores their representations and applications in linear regression and Chebyshev functionals.

## Contribution

It defines the weak n-inner product, relates it to standard k-inner products via Dodgson's identity, and applies it to statistical modeling and functional analysis.

## Key findings

- Representation of n-iterated 2-inner product in terms of standard k-inner products
- Characterization of linear regression models using weak n-inner products
- Generalization of Chebyshev functional with n-iterated 2-inner product

## Abstract

In this article we study a generalization of the n-inner product which we name weak n-inner product. As particular case we consider the n-iterated 2-inner product and we give its representation in terms of the standard k-inner products, k<= n, using the Dodgson's identity for determinants. Finally, we present several applications, including a brief characterization of a linear regression model for the random variables in discrete case and a generalization of the Chebyshev functional using the n-iterated 2-inner product.