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Is there anything wrong with the following:

If $\{f_n\}$, $\{g_n\}$ are two sequences of functions in a Hilbert space $H$, then

$\begin{align*} \sqrt{\sum_n |\langle f,f_n \rangle|^2} - \sqrt{\sum_n |\langle f,f_n - g_n \rangle|^2}&\leq \sqrt{\sum_n |\langle f,g_n \rangle|^2}\\ &\leq \sqrt{\sum_n |\langle f,f_n \rangle|^2} + \sqrt{\sum_n |\langle f,f_n - g_n \rangle|^2} \end{align*}$

for all $f\in H$. Thanks!

1 Answers 1

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This works fine provided that you know all the sums are finite.

An easy way to see it is to let $a(n) = \langle f, f_n \rangle$, $b(n) = \langle f, f_n - g_n \rangle$. Then your inequalities read $||a|| - ||b|| \le ||a-b|| \le ||a|| + ||b||$ where $||\cdot||$ is the $\ell^2$ norm, and this is a simple consequence of the triangle inequality for the $\ell^2$ norm.

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    @Cody: Sure, you could say that. Or, observe that if $||a|| \le ||b||$ then $||a||-||b|| \le 0$, while $||a-b|| \ge 0$ by definition of the norm, so the inequality is trivially satisfied.2012-01-13