[tex]

\mathbf{T}:\overbrace{V\times\cdots\times V}^{n}\times\underbrace{V^{\star}\times \cdots\times V^{\star}}_{m}\longrightarrow\mathbb{R}

[/tex]

I am happy with this definition, it makes a lot of sense to me. However, the physics definition is that of transformations between co-ordinates of the coefficients of the tensor. I can't quite figure out how these two definitions match up.

Any suggestions?