Suppose that in we use an alternate definition of the dot product,

Prove that all of the properties of Theorem 12.2 (page 451 of Apostol) are still valid. Is the Cauchy-Schwarz inequality still valid?

*Proof.* Let , and .

- We compute,
- We compute,
- We compute,
On the other hand,

- We compute,
if .

- If then

Cauchy-Schwarz still holds since we proved the Cauchy-Schwarz inequality using properties (a)-(e). Since these properties still hold, the same proof is valid.