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Here's a homework question:

Let ${u_1, \ldots, u_n}$ be an ONB in $C^n$. Assuming that $n$ is even, compute

$$||u_1 - u_2 + u_3 -\cdots - u_n||$$

I have no idea how to solve this. Can anyone help?


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"ONB" is not a standard abbreviation -- in any case, I have not used it or seen it being used until now. To make the post self-contained, you should perhaps expand it to "orthonormal basis". – Srivatsan Jan 16 '12 at 22:21
I've seen "ONB" before, but without some context would not have immediately realized what it means. – Michael Hardy Jan 16 '12 at 23:04

Let $n=2p$. We have \begin{align*} \lVert u_1-u_2+\cdots -u_n\rVert^2 &=\lVert \sum_{j=1}^pu_{2j-1}-u_{2j}\rVert^2 \\ &=\sum_{j=1}^p\lVert u_{2j-1}-u_{2j}\rVert^2\\ &= \sum_{j=1}^p 2=2p, \end{align*} so $\lVert u_1-u_2+\cdots -u_n\rVert=\sqrt n$.

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You could do this by expanding out $\|x\|^2$, or perhaps you have seen a formula for $\|\sum_j c_j u_j\|$.

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