all possible inner products in $\mathbb R^2$

Question

Suppose $\langle., .\rangle: \mathbb R^2\times \mathbb R^2\to \mathbb R$ is an inner product.

What would be all possible function forms of the inner products, i.e. would all of them have the forms

either $\langle x, y\rangle=ax_1y_1+bx_2y_2$ or $\langle x, y\rangle=ax_1 y_2+b x_2y_1, a,b\in \mathbb R$

or other forms are also possible?

How about $\mathbb R^n$$?$

Answer 1

By definition an inner product $\langle\cdot,\cdot\rangle$ on a real vector space $V$ is a bilinear, symmetric and positive definite form. In the case of $V=\mathbb{R}^n$ all inner products have form $\langle x,y\rangle=x^TAy$, where $A$ is a symmetric $n\times n$ matrix with $n$ positive eigenvalues.

Answer 2

Since the Gram matrix defining the inner product (it has as entry at place $(i,j)$ the inner product of $e_i$ by $e_j$) is symmetric and positive definite (and conversely), you can characterize it with Sylvester's criterion: all principal minors should be positive.

In the case of a symmetric $2\times 2$ matrix, say $$ \begin{bmatrix} a & b \\ b & c \end{bmatrix} $$ this becomes $$ a>0,\qquad ac-b^2>0 $$ For a $3\times 3$ matrix, say $$ \begin{bmatrix} a & b & c \\ b & d & e \\ c & e & f \end{bmatrix} $$ this is $$ a>0,\quad ad-b^2>0,\quad adf+2bce-c^2d-ae^2-b^2f>0 $$