Simultaneous orthogonalization of inner products in infinite-dimensional vector spaces

TL;DR

This paper investigates conditions under which a family of inner products in infinite-dimensional vector spaces can be simultaneously orthogonalized, providing methods to extend degenerate families to nondegenerate ones.

Contribution

It establishes necessary and sufficient conditions for simultaneous orthogonalization of inner products and introduces constructions to extend degenerate families to nondegenerate ones.

Findings

Simultaneous orthogonalization is possible under certain compatibility conditions.

A nondegenerate element plus compatibility ensures orthogonalization.

Methods to add nondegenerate elements to degenerate families are developed.

Abstract

For an arbitrary field $\mathbb{K}$ and a family of inner products in a $\mathbb{K}$-vector space $V$ of arbitrary dimension, we study necessary and sufficient conditions in order to have an orthogonal basis relative to all the inner products. If the family contains a nondegenerate element plus a compatibility condition, then under mild hypotheses the simultaneous orthogonalization can be achieved. So we investigate several constructions whose purpose is to add a nondegenerate element to a degenerate family and we study under what conditions the enlarged family is nondegenerate.