Inner Product: "Replace the positivity condition in the def. of the inner product..."

[janesummer]

Inner Product: "Replace the positivity condition in the def. of the inner product..."

Hey! I have a question in linear algebra! Can you guys help me?
I can´t prove it...

Suppose F = R and V is not equal to {0}. Replace the positivity condition (which states that < v, v > > 0 for all v elements of V) in the definition of the inner product (6.3) with the condition that < v, v > > 0 for some v in V. Show that this change in the definition does not change the set of functions from V × V to R that are the inner products on V.

Is it enough prove only properties of inner product? Like, positivity, additivity and homogeneity in first slot and conjugate symmetry?
The exercise is that:

 

[HallsofIvy]

Hey! I have a question in linear algebra! Can you guys help me?
I can´t prove it...

Suppose F = R and V is not equal to {0}. Replace the positivity condition (which states that < v, v > > 0 for all v elements of V) in the definition of the inner product (6.3) with the condition that < v, v > > 0 for some v in V. Show that this change in the definition does not change the set of functions from V × V to R that are the inner products on V.

Is it enough prove only properties of inner product? Like, positivity, additivity and homogeneity in first slot and conjugate symmetry?
The exercise is that:

The question appears to ask you to show that, if we drop the requirement of positivity we do not get any new "inner products". That is, if a function from pairs of vectors to the real numbers satisfies additivity and homogeneity then it must also satisfy "positivity". The problem I have with that is that it is clearly not true! The function that assigns 0 to any pair of vectors satisfies every thing except positivity.