Inner Product Uniqueness

Question

So an inner product satisfies these properties:

• $ \bf x \cdot \bf y = \bf y \cdot \bf x $
• $ \bf x \cdot (\bf y + \bf z) = \bf x \cdot \bf y + \bf x \cdot \bf z $
• $ \alpha ( \bf x \cdot \bf y) = (\alpha \bf x) \cdot \bf y $
• $ \bf x \cdot \bf x \ge 0\quad \text {and} \quad x\cdot x = 0\iff x = 0$

Now for $R^n$, the inner product is just the dot product. But is the dot product the only map that satisfies these conditions, or can there be others as well? I'm not interested in trivial extensions to the dot product like $\bf x \cdot \bf y = (\sum_i x_i y_i)/k $ where k is just some constant, because this isn't really different.

Is there some totally different function, such that:

$$ f(\bf x, \bf y) = k $$

Where x and y are vectors and k is a constant, that satisfies all of those properties? Or are those four conditions enough to force us to adopt this form in $R^n$?

Answer 1

The last item should be

• $\mathbf{x}.\mathbf{x}\geqslant 0$ and $\mathbf{x}.\mathbf{x}=0\iff\mathbf{x}=0$.

Yes, there are other inner products. For instance, in $\mathbb{R}^2$ you can define$$(x_1,x_2).(y_1,y_2)=2x_1y_1-x_1y_2-x_2y_1+2x_2y_2.$$