Instead of the usual definition of the dot product, suppose we define the dot product of and by the formula

Which of the properties of Theorem 12.2 (on page 451 of Apostol) are still valid under this new definition? Is the Cauchy-Schwarz inequality still valid?

- Property (a) is still valid since
- Property (b) fails since
while

But, is not necessarily equal to .

- Property (c) is still valid since
- Property (d) is still valid since
if .

- Property (e) is still valid since
if .

- The Cauchy-Schwartz inequality still holds since the formula is identical after squaring.

c does not hold for a negative constant

I agree.