# Confusion over this definition of a tensor

A tensor is defined as an object that linearly maps an ordered pair of vectors from the Cartesian product $V\times V$ to scalars, where both vectors in the ordered pair belong to $V$. Hence
\begin{equation}\tag{1}
\bf{T}: \it{V\times V \rightarrow} \mathbb{R}
\end{equation}
Let $v$ and $w$ be vectors that belong to $V$ and $(v,w)$ belong to $V\times V$. Then
$$\bf{T} \it{(v,w)}=a$$
Now if we represent $\bf{T}$ as the *tensor product* of dual space vectors, $v^*\otimes w^*$ then the above statement is written as (this equation is the definition of tenosr product)
$$\langle v^*,v\rangle \langle w^*,w\rangle =a$$
Hence $a$ is simply $v^*_\mu v^\mu w^*_\nu w^\nu$. The representation of tensor as the tensor product of dual space vectors helps us express this map in terms known objects. Hence a tensor is expressed as
$$\bf{T}=\it{v^*_\mu w^*_\nu e^\mu \otimes e^\nu}$$
Equation $(1)$ is what the author meant when he said that tensor eats a bunch of vectors and spits out a scalar.