Question 1
Let $V,W$ be normed spaces and $\iota:V\times W \rightarrow V\otimes W$ be the canonical (algebraic) bilinear map.
It can be easily shown that for any normed space $X$ and a continuous bilinear map $\phi:V\times W\rightarrow X$, there exists a unique continuous linear transformation $\psi:V\otimes_\pi W\rightarrow X$ satisfying $\psi\circ \iota=\phi$.
Hence, $(V\otimes_\pi W,\iota)$ solves this universal mapping problem.
Does $(V\otimes_\epsilon W,\iota)$ also solve this universal mapping problem? If so, then $id:V\otimes_{\epsilon} W\rightarrow V\otimes_{\pi} W$ must be a linear homeomorphism and this means that $\epsilon$-norm and $\pi$-norm are equivalent. Are they?
Question 2
Say, the answer for the question $1$ is no. Then, what's the importance of $\epsilon$-norm? Is there any functorial property related to $\epsilon$-norm?
Question 3
Let $V$ be a normed space and define $T^{r,s}(V):=V\otimes \cdots \otimes V \otimes V^*\otimes \cdots \otimes V^*$ be the tensor product of type $(r,s)$. (Here, $V^*$ is the topological dual of $V$)
Let $\mathscr{T}(V):=\oplus_{r,s\in\mathbb{N}} T^{r,s}(V)$ be the tensor algebra. Then, does there exists a natural topology $\tau$ on $\mathscr{T}$ satisfying the following?
(i) the subspace topology on $T^{r,s}(V)$ induced by $\tau$ is the topology generated by the $\pi$-norm (that is, the subspace topology is the same as the topology on $V\otimes_\pi \cdots \otimes_\pi V \otimes_\pi V^*\otimes_\pi \cdots \otimes_\pi V^*$)
(ii) $\tau$ is normable
(iii) the multiplication operator $\mathscr{T}(V)\times \mathscr{T}(V)\rightarrow \mathscr{T}(V)$ is continuous
More generally, is there a natural norm on graded algebra of normed spaces so that each component becomes a subspace?
Thank you in advance!
