Describe explicitly all inner products on $\mathbb{R}$ and $\mathbb{C}$

Question

I know this is an elementary question, however I am really lost as to where to start.

Since both $\mathbb{R}$ and $\mathbb{C}$ are finite-dimensional I think the inner product will be completely determined by the basis $\{1\}$.

I am not sure where to go from here.

Answer 1

It seems the following.

We shall consider these fields as vector spaces over itself.

Let $x,y\in\Bbb R$. Then $(x,y)=xy(1,1)$, so an inner product on $\Bbb R$ is completely determined by the value $(1,1)$. Conversely, it is easy to check that for each $c>0$ the function $f_c:\Bbb R\times \Bbb R$, $f(x,y)=xyc$ is an inner product at the space $\Bbb R$.

Let $x,y\in\Bbb C$. Then $(x,y)=x\bar y(1,1)$, so an inner product on $\Bbb C$ is completely determined by the value $(1,1)$. Conversely, it is easy to check that for each $c>0$ the function $f_c:\Bbb C\times \Bbb C$, $f(x,y)=x\bar yc$ is an inner product at the space $\Bbb C$.