# Improving the Cauchy-Schwarz inequality

## TL;DR

This paper explores an improved version of the Cauchy-Schwarz inequality, demonstrating its advantages in singular cases and applications, and highlighting the role of projections in enhancing inequality relations.

## Contribution

It introduces a variant of the Cauchy-Schwarz inequality that performs better in singular situations and discusses the impact of projection operators on these inequalities.

## Key findings

- The variant works effectively in singular cases where the original inequality fails.
- Applications show improved bounds in uncertainty relations.
- Projection operators can modify and enhance the inequality relations.

## Abstract

We highlight overlap as one of the simplest inequalities in linear space that yields a number of useful results. One obtains the Cauchy-Schwarz inequality as a special case. More importantly, a variant of it is seen to work desirably in certain singular situations where the celebrated inequality appears to be useless. The basic tenet generates a few other interesting relations, including the improvements over certain common uncertainty bounds. Role of projection operators in modifying the Cauchy-Schwarz relation is noted. Selected applications reveal the efficacy.

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Source: https://tomesphere.com/paper/1907.05024