# Tensor of type (k,l)

1. Feb 9, 2010

### Fredrik

Staff Emeritus
In John Lee's books "Introduction to smooth manifolds" and "Riemannian manifolds", a tensor of type $$\begin{pmatrix}k\\ l\end{pmatrix}$$ on a vector space V is defined as a member of

$$\underbrace{V^*\otimes\cdots\otimes V^*}_{k}\otimes\underbrace{V\otimes\cdots\otimes V}_{l}$$

or as a multilinear function

$$\underbrace{V^*\times\cdots\times V^*}_{l}\times\underbrace{V\times\cdots\times V}_{k}\rightarrow\mathbb R$$

or, when $l>0$, as a multilinear function

$$\underbrace{V^*\times\cdots\times V^*}_{l-1}\times\underbrace{V\times\cdots\times V}_{k}\rightarrow V$$

(These three vector spaces are isomorphic). But in Wald's "General relativity", this is called a tensor of type $(l,k)$. I just want to ask, is this a "math vs. physics" thing, like when physicsts make their inner products antilinear in the first variable and mathematicians make theirs antilinear in the second? Or is there a standard convention that one of these guys is ignoring?

2. Feb 18, 2010

### enricfemi

i saw the type of (l,k) for both of math and physics.