Three kinds of tensor

I've written before about how the word tensor" is applied to several different things, and that the connection between them isn't immediately obvious. I thought about writing more about that some day, but I recently became aware of a paper that does this more thoroughly than I ever could.

The paper looks at three notions of a tensor

1. a multi-indexed object that satisfies certain transformation rules
2. a multilinear map
3. an element of a tensor product of vector spaces

and explores (for 210 pages!) how they are related. The length comes primarily from the number of examples. The author is pulling on a thread that runs throughout a lot of mathematics.

Here's the paper:

Lek-Heng Lim. Tensors in computations. Acta Numerica (2021), pp. 1-210. doi:10.1017/S0962492921000076. Available here.

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