I've been trying to figure this homework problem out for the last hour and can't seem to get anything going. Looking for some hints or help.
I've been trying to start using the Cauchy-Schwarz Inequality and work from there.
I've also tried, for example, something like: $\sum_{i=1}^n(\frac{x \cdot v_i}{v_i \cdot v_i})^2 \leq x \cdot x$ and then trying to use the fact that $|x \cdot v_i| \leq ||x||\cdot||v_i||$, but that still didn't yield much.
I also tried replacing $x$ in $<x,x>$ with $\sum_{i=1}^nc_iv_i$ (because $x$ is a linear combination) but still I don't know where to go from here...
What's a good starting point? Can you point me in the right direction?
EDIT: Here's what I've been trying...
I started with the Fourier Trick and worked from there:
$$c_i=x\cdot v_i$$ $$c_i \leq ||x||\cdot||v_i||$$ $$c_i \leq ||x||$$ (since the basis is orthonormal) $$c_i^2 \leq ||x||^2$$ $$\sum_{i=1}^nc_i^2 \leq n||x||^2$$ $$\frac{1}{n}\sum_{i=1}^nc_i^2 \leq ||x||^2$$
It seems like I'm getting closer, but this still isn't quite the answer they're look for.
Write $x\in V$ as an orthogonal decomposition: $$ x = \sum_{k=1}^{n}\langle x,c_k\rangle c_k+\left(x-\sum_{k=1}^{n}\langle x,c_k\rangle c_k\right). $$ Conclude that \begin{align} \|x\|^2 &= \left\|\sum_{k=1}^{n}\langle x,c_k\rangle c_k\right\|^2+\left\|x-\sum_{k=1}^{n}\langle x,c_k\rangle c_k\right\|^2 \\ &\ge \left\|\sum_{k=1}^{n}\langle x,c_k\rangle c_k\right\|^2\\&=\sum_{k=1}^{n}|\langle x,c_k\rangle|^2. \end{align}