Suppose $H_1$ and $H_2$ are two Hilbert spaces with dimension $n$ and $m$, for $ x \in H_1 \otimes H_2$

consider $$\|x\|_\pi = \inf \left\{ \sum_{i=1}^n \|a_i\| \|b_i\| : x = \sum_{i} a_i \otimes b_i \right\}$$ where $x \in A \otimes B$.

$ \|a\otimes b\|_{2}^2 =\langle a \otimes b,a\otimes b\rangle = \langle a,a\rangle_1 \, \langle b,b\rangle_2 \quad \mbox{for all } a \in H_1 \mbox{ and } b \in H_2 $ respectively.

**Questions:**
1. We know norms on finite dimensional spaces are equivalent. Hence there are $c_1$ and $c_2$ such that $$
c_1\|x\|_2
\leq \|x\|_\pi
\leq c_2\|x\|_2
$$ since $\|x\|_\pi$ is largest cross norm so $c_1$ can be taken equal to $1$. what about $c_2$

2.What about when $H_1$ and $H_2$ are arbitrary Hilbert spaces, are these two norm equivalent?

- Where can I know more about such comparing norms on tensor of Hilbert spaces. (any book or article)

thanks, and pardon me for poor English.