# The tensor algebra

We first use the tensor product to generate an algebra from an $${n}$$-dimensional vector space $${V}$$. The $${k^{\textrm{th}}}$$ tensor power of $${V}$$, denoted $${T^{k}V}$$, is the tensor product of $${V}$$ with itself $${k}$$ times; it forms a vector space of dimension $${n^{k}}$$. The tensor algebra of $${V}$$ is then defined to be the infinite direct sum of every tensor power:

$$\displaystyle TV\equiv\sum T^{k}V=\mathbb{R}\oplus V\oplus(V\otimes V)\oplus(V\otimes V\otimes V)\oplus\dotsb$$

(however, see the last paragraph of the section on Tensors).

The vector multiplication operation is $${\otimes}$$, and thus the infinite-dimensional tensor algebra is associative. In fact, the tensor algebra can alternatively be defined as the free associative algebra on $${V}$$, with juxtaposition indicated by the tensor product.

If a pseudo inner product, i.e. a nondegenerate symmetric bilinear form, is defined on $${V}$$, it can be naturally extended to any $${T^{k}V}$$ by extending the pairwise operation defined previously: if $${A=v_{1}\otimes v_{2}\otimes\dotsb\otimes v_{k},}$$ and $${B=w_{1}\otimes w_{2}\otimes\dotsb\otimes w_{k}}$$, we define

$$\displaystyle \left\langle A,B\right\rangle \equiv\prod\left\langle v_{i},w_{i}\right\rangle .$$

The pseudo inner product can then be extended to a nondegenerate symmetric multilinear form on all of $${TV}$$ by defining it to be zero between elements from different tensor powers.