The exact constant for the $\ell_1-\ell_2$ norm inequality

TL;DR

This paper provides a method to compute the exact constant in the $\,\ell_1-\ell_2$ norm inequality for any vector in Hilbert space, refining a fundamental inequality with broad theoretical and practical implications.

Contribution

It introduces a trivial method to determine the exact constant for the $\,\ell_1-\ell_2$ inequality for each vector, enhancing the understanding of this key Hilbert space inequality.

Findings

Exact constants can be computed for each vector.

The method refines the classical inequality.

Potential for widespread applications in Hilbert space theory.

Abstract

A fundamental inequality for Hilbert spaces is the $\ell_1-\ell_2$-norm inequality which gives that for any $x \in H^n$, $\|x\|_1\le \sqrt{n}\|x\|_2.$ But this is a strict inequality for all but vectors with constant modulus for their coefficients. We will give a trivial method to compute, for each x, the constant $c$ for which $\|x\|_1=c\sqrt{n}\|x\|_2.$ Since this inequality is one of the most used results in Hilbert space theory, we believe this will have unlimited applications in the field. We will also show some variations of this result.