Scalar / Dot product

Question

I have a simple question about Scalar / Dot product. (http://en.wikipedia.org/wiki/Dot_product)

Say f is a bilinear form. I have to tell if f defines a dot product. I didn't understand what I should do, what does f has to satisfy so it can be called a "dot product" ?

thanks

Answer 1

Let $V$ be a vector space over $\Bbb R$.

A bilinear form $f:V\times V\to\Bbb R$ is a scalar product if it satisfies

1. $f(x,y)=f(y,x)$ for all $x,y\in V$ (symmetric)
2. $f(x,x)> 0$ for all nonzero $x\in V$ (positive definite)