Tensor algebra

Remarks

$T(V)$ is an associative unital algebra over $F$, graded by the degree $k$, and $V = T^1(V)$ sits inside it. Universal property: for every associative unital $F$-algebra $A$ and every linear map $f \colon V \to A$, there is exactly one unital algebra homomorphism $\bar f \colon T(V) \to A$ with $\bar f(v) = f(v)$ for $v \in V$. Example: if $V = F x$ has dimension $1$, then $T^k(V) = F x^{\otimes k}$, and $x^{\otimes k} \mapsto x^k$ is an isomorphism $T(V) \cong F[x]$; the check is that both products add degrees. Non-example: for $V = F^2$ with basis $x, y$, the polynomial ring $F[x, y]$ is not $T(V)$, because $x \otimes y \ne y \otimes x$ in $T(V)$ while $xy = yx$ in $F[x, y]$. If $V$ has a basis $(e_i)_{i \in I}$, then $T(V)$ is the free associative algebra on the set $I$. The same construction works for a module over a commutative ring.