# 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.