What is the Cauchy-Schwarz Inequality?

Question

So I've been thinking of doing some Olympiad math (USAMO, USAJMO) in the future and am trying to prepare for it. One concept I don't quite understand is the Cauchy-Schwarz Inequality. Most likely I will need to use it in sums if I approach it. I know that there are many forms of the Cauchy-Schwarz Inequality, but what is the sum form of it?

I am in middle school, but I will still understand sums written as $\sum$.

Answer 1

According to Wikipedia:

The Cauchy–Schwarz inequality states that for all vectors $u$ and $v$ of an inner product space it is true that $$\left|\langle u,v\rangle\right|\leq\left\|u\right\|\left\|v\right\|.$$

If, in particular, the inner product space is $\mathbb C^n$, which has inner product

$$\langle u,v\rangle=\langle\left(u_1,u_2,\dots,u_n\right),\left(v_1,v_2,\dots,v_n\right)\rangle=\sum_{k=1}^nu_k\bar v_k,$$

then the above inequality becomes

$$\left|\langle u,v\rangle\right|=\left|\sum_{k=1}^nu_k\bar v_k\right|\leq\sqrt{\sum_{k=1}^n\left|u_k\right|^2}\sqrt{\sum_{k=1}^n\left|v_k\right|^2}=\left\|u\right\|\left\|v\right\|.$$