This is a question of the book Linear Algebra of Kenneth Hoffman.
Describe explicitly all inner products on $\mathbb{R}$ and on $\mathbb{C}$.
I think that if $<,>$ is one inner product on $\mathbb{R}$ (or $\mathbb{C}$) then $< ,>=K[,]$, where $[,]$ is the standard inner product. Is that correct?
Yes. If $x\in\mathbb{R}$ then $x=x\cdot1$ is how $x$ is written in the basis $\{1\}$ of $\mathbb{R}$, where $\cdot$ is the multiplication by an scalar. Therefore $(x,y)=(x\cdot1,y\cdot1)=xy(1,1)$. Therefore $(x,y)=(1,1)[x,y]$, in your notation $[x,y]=xy$. So, you $K$ is the number $(1,1)$.
Exactly the same explanation works for $\mathbb{C}$ except that the $y$ comes out of the scalar product as $\overline{y}$. So, we get $(x,y)=x\overline{y}(1,1)$. But $[x,y]=x\overline{y}$.
More care is needed when $\mathbb{C}$ is being considered as a vector space over $\mathbb{R}$, i.e. the scalars are only the real numbers.
