n-fold Tensor Power

linear-algebra

Definition

$n$-fold Tensor Power

Let $V\cong F^n$ be a vector space over a field $F$.

The $n$-fold tensor power of $V$, denoted as $V^{\otimes n}$ is defined recursively by:

$$\begin{array}{rl}{V}^{\otimes 0}& :=F\\ V^{\otimes 1}& :=V\\ V^{\otimes n}& :=V\otimes V^{\otimes (n-1)}\forall n\ge 2\end{array}$$

where $\otimes$ denotes the tensor product.