In physics tensor fields are often defined in a different way than in math. I am trying to establish an equivalence between the definitions.
In math a tensor field of type $(k,l)$ on a manifold $M$ is an assignment of a tensor of type $(k,l)$ on the tangent space $T_p M$ for every $p\in M$. A tensor of type $(k,l)$ on a vector space $V$ being a multilinear function $(V^*)^k \times V^l\to \mathbb{R}$.
In physics a tensor field is often defined by a transformation law. If $(u^i)_i$ and $(u'^i)_i$ are two local coordinate systems on coordinate neighbourhoods $U$ and $U'$ of $M$ respectively, a tensor field $T$ on $M$ is defined as an object whose components transform as: $$T'\hphantom{'}^{i_1 \dots i_k}_{j_1 \dots j_l} = \frac{\partial u'^{i_1}}{\partial u^{a_1}} \dots \frac{\partial u'^{i_k}}{\partial u^{a_k}} \frac{\partial u^{b_1}}{\partial u'^{j_1}} \dots \frac{\partial u^{b_1}}{\partial u'^{j_l}} T^{a_1 \dots a_k}_{b_1 \dots b_l}$$
The only way to make sense of this definition is, I think, to let T be an assignment of a map $((T_p M)^*)^k \times (T_p M)^l\to \mathbb{R}$ to every point $p\in M$, but not necessarily a multilinear one.
Due to the multilinearity the math definition implies the physics definition immediately. What's left to prove is thus that the maps from the physics definition are multilinear.
My idea was the following. Let $(u^i)_i$ be a local coordinate system and let $(e_i)_i$ be the associated basis of $T_p M$ and $(\epsilon ^i)_i$ its dual basis. Suppose T is a $(1,1)$ tensor field from the physics definition and suppose we want to show $T(\epsilon ^2 , 2 e_1)=2T(\epsilon ^2 , e_1)$. Then we can define a new coordinates by $u'^1=\frac{1}{2}u^1$ and $u'^i= u^i$ for all other $i$. Then we get: $$T(\epsilon ^2 , 2 e_1) = T(\epsilon' ^2 , e'_1) = T'^2_1 = \frac{\partial u'^2}{\partial u^a}\frac{\partial u^b}{\partial u'^1}T^a_b = \frac{\partial u'^2}{\partial u^2}\frac{\partial u^1}{\partial u'^1}T^2_1 = 2T^2_1 = 2T(\epsilon ^2 , e_1)$$ and we are done.
The only problem is that this approach doesn't work when instead of $T^2_1$ we would look at e.g. $T^1_1$, since the dual basis vector used then transforms along with the basis vector. So when $2e_1=e'_1$ then $\epsilon_1=2\epsilon'_1$ and the trick cannot be used.
Have I understood the physics defintion correctly and if so, is there a way to work around this problem?