Suppose that $V$ is a finite n-dimensional Banach space, and suppose that $T(V)$ is the tensor algebra on it. Furthermore, suppose that $T^{(n)}(V)$ is the "abridged" tensor algebra obtained by taking the direct sum of all up to the nth tensor power. This is to keep everything nice and finite-dimensional.
Grothendieck's projective norm on $T^{(n)}(V)$ for an arbitrary element $v$ in this algebra is
$\|v\|_\wedge = \inf \left\{ \sum_{i\leq n} \left( \prod_{j \leq i}\|v_{ij}\| \right)\right\}$
where the $v_{ij}$ are simple tensors of rank 1, and where $i$ represents the order of the tensor, and where the infimum is taken over all possible representations of $v$.
Question: Does Grothendieck's injective norm also work out nicely to
$\|v\|_\vee = \inf \left\{ max\left( \prod_{j \leq i}\|v_{ij}\| \right)\right\}$
?
I'm aware that the injective norm is usually not defined this way - rather, it's more often defined as a certain dual norm (which I won't rehash here). In fact, the projective norm is also typically defined as a certain dual norm, but it's well-known that the first expression given above is equivalent to it. I'm simply curious if my proposed second expression is equivalent to the injective norm in the same way.
I'm curious if it works out, either for $T^{(n)}(V)$ or for $T(V)$.