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P.G. Casazza, F. Philipp and G. Kutyniok, An Introduction to Finite Frame Theory
2012
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P.G. Casazza and R. Lynch A brief introduction to Hilbert space frame theory and its applications
2016
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2017

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

Botelho-Andrade, Sara, Casazza, Peter G., Cheng, Desai et al.

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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.