Multiplication in symmetric product space STATEMENT: Let $V=\mathbb{R}^2$.Take $Y:=\left\{x\cdot y: x,y\in V\right\}$ where $S_2(V)$ is the symmetric product of $V$.
QUESTION: What is multiplication in the symmetric product space?

The symmetric power $S^p(V)$ of a vector space $V$ is defined as the quotient $V^{\otimes p} / (v_1 \otimes \dotsc \otimes vp = v_{\sigma(1)} \otimes \dotsc \otimes v_{\sigma(p)} : v_i \in V, \sigma \in \Sigma_p)$. The natural isomorphism $V^{\otimes p} \otimes V^{\otimes q} \to V^{\otimes p+q}$ extends to a linear map $S^p(V) \otimes S^q(V) \to S^{p+q}(V)$. It maps $[v_1 \otimes \dotsc \otimes v_p] \otimes [w_1 \otimes \dotsc \otimes w_q]$ to $[v_1 \otimes \dotsc \otimes v_p \otimes w_1 \otimes \dotsc \otimes w_q]$. Usually one writes $[v_1 \otimes \dotsc \otimes v_p]$ in $S^p(V)$ as $v_1 \cdot \dotsc \cdot v_p$, so that $S^p(V) \otimes S^q(V) \to S^{p+q}(V)$ is really just a multiplication. In your question, $p=q=1$. Hence, $S^2(V) = V^{\otimes 2}/(v \otimes w = w \otimes v)$, where $[v \otimes w]$ is denoted by $v \cdot w$ (which equals $w \cdot v$).
