Free associative algebra on a vector space

Vector spaces and associative algebras will be implicitely all on the same field \K = \R or \C.

Let U be a vector space. We wish to build a free associative algebra V on U.

For any natural number n \ge 1, we will note U^{\otimes n} = U \otimes U \otimes \ldots \otimes U (tensor product n times).

Each U^{\otimes n} is a vector space. We may form:

    \[V = U \oplus U^{\otimes 2}  \oplus U^{\otimes 3} \ldots = \bigoplus_{n \in \N_1} {U^{\otimes n}}\]

An element \vec v of V can be written (\vec v_1, \vec v_2, \ldots \vec v_i, \ldots) with each \vec v_i element of U^{\otimes i}, and \vec v_i being non null for only a finite number of indices i.