- #1
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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?
\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?