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.