I am working through linear algebra problems in Apostol's Calculus, and he has numerous problems that seem to imply that Cauchy-Schwarz holds no matter how the inner product is defined. Then, he has problems where the triangle inequality holds despite alternative definitions of vector norm. This got me thinking, since the proof of the triangle inequality in Apostol relies on Cauchy-Schwarz, that the triangle inequality would hold regardless of how the vector norm is defined (if it involves the dot product).
I then found this response to a question, which states what I was thinking.
Are there any proofs that the Cauchy-Schwarz inequality holds in any inner product space (I looked for some and found none and could not prove it myself)? I've had a semester of algebra (Artin) and some analysis, if there is a proof at such a level of understanding. Intuitive explanations are good, too.