I am looking to understand why the free tensor series, denoted $T((V))$, is not an algebra, while the free tensor algebra, denoted $T(V)$, is.
To clarify definitions:
- Let $V$ be a vector space over a field $F$.
- The tensor algebra $T(V)$ is the direct sum of tensor powers of $V$, i.e., $T(V) = \bigoplus_{n=0}^{\infty} T^n(V)$, where $T^n(V) = V^{\otimes n}$.
- The tensor series $T((V))$ is the set of all sequences $(a_0, a_1, a_2, \ldots)$ where $a_i \in T^i(V)$.
I understand that $T(V)$ is an algebra because it is closed under the operations of vector addition and tensor multiplication, both of which are well-defined.
The main problem I see for $T((V))$ is that multiplication might not be well-defined. Let $A = (a_0, a_1, a_2, \ldots)$ and $B = (b_0, b_1, b_2, \ldots)$ be two elements of $T((V))$. When we try to define the product $AB = (c_0, c_1, c_2, \ldots)$, the coefficient $c_n$ is the sum over all $i$ from $0$ to $n$ of $a_i \otimes b_{n-i}$.
To further simplify the question, consider the case where $V$ is a one-dimensional vector space over $F$. In this case, the tensor algebra $T(V)$ is isomorphic to the polynomial ring $F[x]$, and the tensor series $T((V))$ is isomorphic to the ring of formal power series $F[[x]]$. If I understand correctly, $F[[x]]$ is an algebra.
Why is it that $T((V))$ fails to be an algebra in general, whereas $F[[x]]$ is an algebra? I would appreciate any insights or references that could help shed light on this question.