# Tensors as multilinear mappings

There is an obvious multiplication of two 1-forms: the scalar multiplication of their values. The resulting object $${\varphi\psi\colon V\times V\to\mathbb{R}}$$ is a nondegenerate bilinear form on $${V}$$. Viewed as an “outer product” on $${V^{*}}$$, multiplication is trivially seen to be a bilinear operation, i.e. $${a\left(\varphi+\psi\right)\xi=a\varphi\xi+a\psi\xi}$$. Thus the product of two 1-forms is isomorphic to their tensor product.

We can extend this to any tensor by viewing vectors as linear mappings on 1-forms, and forming the isomorphism $${\bigotimes\varphi_{i}\mapsto\prod\varphi_{i}}$$. Note that this isomorphism is not unique, since for example any real multiple of the product would yield a multilinear form as well. However it is canonical, since the choice does not impose any additional structure, and is also consistent with considering scalars as tensors of type $${\left(0,0\right)}$$.

We can thus consider tensors to be multilinear mappings on $${V^{*}}$$ and $${V}$$. In fact, we can view a tensor of type $${\left(m,n\right)}$$ as a mapping from $${i<m}$$ 1-forms and $${j<n}$$ vectors to the remaining $${\left(m-i\right)}$$ vectors and $${\left(n-j\right)}$$ 1-forms. The above shows different ways of depicting a pure tensor of type $${\left(1,2\right)}$$. The first line explicitly shows the tensor as a mapping from a 1-form $${\varphi}$$ and a vector $${v}$$ to a 1-form $${\xi}$$. The second line visualizes vectors as arrows, and 1-forms as receptacles that when matched to an arrow yield a scalar. The third line combines the constituent vectors and 1-forms of the tensor into a single symbol $${T}$$ while merging the scalars into $${\xi}$$ to define $${\zeta}$$, and the last line adds indices (covered in the next section).

A general tensor is a sum of pure tensors, so for example a tensor of the form $${\left(u\otimes\varphi\right)+\left(v\otimes\psi\right)}$$ can be viewed as a linear mapping that takes $${\xi}$$ and $${w}$$ to the scalar $${\xi\left(u\right)\cdot\varphi\left(w\right)+\xi\left(v\right)\cdot\psi\left(w\right)}$$. Since the roles of mappings and arguments can be reversed, we can simplify things further by viewing the arguments of a tensor as another tensor: $${\left(u\otimes\varphi\right)\left(\xi\otimes w\right)\equiv\left(u\otimes\varphi\right)\left(\xi,w\right)=\left(\xi\otimes w\right)\left(u,\varphi\right)=\xi\left(u\right)\cdot\varphi\left(w\right)}$$.