Polynomial in a $2$-dimensional Euclidean space

Question

I am reading about the problem of moments and get some troubles in they way one defines the measure on some $k$-dimensional Euclidean space by polynomial. Could you please help me?

More precisely, I am reading the article "The Problem of Moments", written by Shohat in 1943. In this article, the author let $R$ be a $2$ - dimensional vector space. The author also put $P (u,v)$ as any polynomial in $u,v \in R$. $$P(u,v) = \sum_{i,j} x_iy_j u^i v^j,$$ where $x_i, y_j$ are real or complex - valued constants.

After that, the author introduced the functional $\mu (P) = \sum_{i,j} \mu_{i,j} x_i y_j$.

So, my questions are the following.

1. How can we define $u^i, v^j$ in $R$, where $u$ and $v$ are some vectors in $R$?

2. What is the definition of a polynomial in a $2$-dimensional vector space? I tried to search on the internet, however, no useful material has been found.

3. What is $\mu_{ij}$ in his formula?