Does there exist an inner product on $\mathbb R$ besides the canonical one?

Question

Let $f(x,y)$ be an inner product on the $\mathbb R$-vector space $\mathbb R$.

Is it true that $f$ must be of the form $cxy$ for some nonnegative constant $c$?

Answer 1

Yes because the conditions

• $\langle \alpha x,y\rangle=\alpha\langle x,y\rangle$
• $\langle x,y\rangle=\langle y,x\rangle$
• $\langle x,x\rangle\geq 0$

imply

• $f(x,y)=xyf(1,1)$
• $f(1,1)\geq 0$

So, $f(x,y)=cxy$ for the nonnegative constant $c=f(1,1)$.

Answer 2

To complement @Pedro's answer, we note that $f(x,y)$ is an inner product on $\Bbb R^{n}$ if and only if there exists a positive definite matrix $M\in\Bbb R^{n\times n}$ such that $f(x,y)=x^TMy$ for every $x,y$. In the particular case of $n=1$, we have $M\in\Bbb R$ is positive definite if and only if $M>0$, that is $f(x,y)=cxy$ with $c=M$.