1

Let $X$ be a Hilbert space, $x \in X$, $\{e_i\}$ a countably infinite orthonormal basis.

I think I remember seeing in a book that $$ \sum_{i} |\langle x,e_i \rangle| \leq 4 \left( \sum_{i} |\langle x,e_i \rangle |^2 \right)^{1/2} $$ (or something like that). But I can't find this inequality anywhere and can't seem to prove it.

How can it be proved or disproved?

By Parseval, the right-hand side is $4\| x \|_2$.

I know the inequality $$ \sum_{i} |c_i| \leq C \left( \sum_{i} |c_i|^2 \right)^{1/2} $$ cannot hold in general (See the answer to $\ell^p\subseteq\ell^q$ for $0<p<q<\infty$ and $\|\cdot\|_p<\|\cdot\|_q$ )

But maybe there is some orthogonality or parallelogram law trick that works when $c_i = \langle x,e_i \rangle$.

RitterSport
  • 1,583

1 Answers1

2

There can be no such inequality. To see this, let $N$ be a positive integer and let $$ x = \sum_{j=1}^N \frac{1}{j} e_j. $$ Then $$ \ln(N+1) < \sum_{j=1}^N |\langle x, e_j \rangle| $$ but $$ \sum_{j=1}^N |\langle x, e_j \rangle|^2 < \frac{\pi^2}{6}. $$ Thus the left side of the alleged inequality gets arbitrarily large as $N$ gets large, but the right side stays bounded.

Sheldon Axler
  • 7,056