Let V be a vector space and < , > be a positive definite inner product and B = { , ..., } be a basis for V. If , ..., in K are scalars, then show that there is a unique v in V such that < v, > = .

This is what I have tried so far.

Since we have a basis for V, we can write v as a linear combination of the basis elements.

v = + + ... + + ... +

< v, > = ( + + ... + + ... + ) *

< v, > = + + ... + + ... + we want all of this to equal

So this is where I am stuck. Somehow I want everything to drop off but . If this was an orthogonal basis, all the , where j is not equal to i, would drop out but it is a regular basis.

Any help would be appreciated.