If we define the dot product by the formula

prove that the properties of the dot product in Theorem 12.2 (page 451 of Apostol) continue to be valid.

*Proof.* Let , and .

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

- We compute,
if .

- If then

Again, Cauchy-Schwartz will hold since the proof of Cauchy-Schwarz relied only on properties (a)-(e), which are all still valid under this new definition of the dot product.