Note that a 2-homogeneous polynomial on Banach space $X$ is the mapping $p:X\rightarrow\mathbb{C}$ such that $p(x)=A(x,x)$ where $A:X\times X\rightarrow\mathbb{C}$ is a bilinear form on $X$.
The norm of this polynomial is $\sup_{\Vert x\Vert=1}\Vert p(x)\Vert$.
I studied that every norm-attaining 2-homogeneous polynomial on $c_0$ is in fact finite, which means there is $N$ such that $p((x_1,x_2,...,x_n,...))=p((x_1,x_2,...,x_N,0,0,0...))$. So this is a 2-homogeneous polynomial on $l_\infty^n$.
However, I don't understand why then there is a 'unique' Hahn-Banach extension of $p:l_\infty^n\rightarrow\mathbb{C}$ into $\hat{p}:l_\infty\rightarrow\mathbb{C}$.($l_\infty^n$ is the finite dimensional space with maximum norm, $l_\infty$ means a space of sequences whos supremum is bounded) Which is the 2-homogeneous polynomial on $l_\infty$ that extends $p$ and preserves its norm. Is there any theory about $l_\infty^n$ related to this? Thanks for any helps.