Intuition for the Cauchy-Schwarz inequality that does not rely on a geometric interpretation of vectors I know that the Cauchy-Schwarz inequality can be understood by interpreting vectors as arrows in the coordinate plane, but since the inequality should hold for any inner product space, I'm looking for an intuitive explanation for why the inequality should be true based only on the abstract definitions of a vector space and the inner product. I understand that the inequality can be proven mathematically without relying on geometry, but the proofs I've seen feel arbitrary and unintuitive.
 A: I'm not going to provide you with the insight you are looking for, nor am I certain whether that's possible.
Clearly the Cauchy-Schwarz inequality stems form Euclidean geometry. The expression $\left\langle \frac{x}{\|x\|},\frac{y}{\|y\|}\right\rangle$ is roughly the projection of the unit vector $\frac{y}{\|y\|}$ onto the direction of $\frac{x}{\|x\|}$. In that sense it's perfectly sensible that $$\left|\left\langle \frac{x}{\|x\|},\frac{y}{\|y\|}\right\rangle\right|\leq 1.$$
This explanation to me justifies the inequality and provides roughly all the intuition I have about this inequality.
The fact that this inequality makes sense in the complex space $\mathbb{C}^n$ is already baffling and difficult to understand intuitively. On the other hand, the inequality is fairly easy to prove for general inner product spaces. For me, the latter is exactly why we study mathematics in the first place. Mathematics is simply a language that provides powerful notations, techniques and tools to understand these results and being able to use them.
If someone is able to provide general intuition that would be awesome!
