On page 71 of The Absolute Differential Calculus by Levi-Civita, a very clear definition of a tensor is given in terms of how the coefficients of a multi-linear form transform, such that the product with its dual multi-linear form remains invariant:

An m-fold covariant is an m-fold system which is transformed in the same way as the coefficients of a multilinear form in point variables; an m-fold contravariant is one which is transformed in the same way as the coefficients of a multilinear form in dual variables; more generally, a mixed system or tensor is one which is transformed in the same way as the coefficients of a multilinear form in both point and dual variables (including also as particular cases both purely covariant and purely contravariant systems).

Were there other related definitions defined around this time?

How has today's definition of a tensor changed compared to its beginnings?

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    $\begingroup$ I think at some point physicists and mathematicians went different ways on this. Physicists still prefer to define things by their transformation properties. I think mathematicians normally prefer coordinate-independent descriptions of this kind of thing, so they wouldn't define a tensor in terms of how its coefficients transform. $\endgroup$ – Ben Crowell Jun 17 '15 at 3:32
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    $\begingroup$ It isn't clear to me that "today's definition of a tensor" exists uniquely, even if we restrict to pure mathematics. For instance, while the definition one learns in elementary linear algebra is in terms of bilinear maps, I've known mathematicians who would rather define it based on, for example, tensor-hom adjunction. $\endgroup$ – Logan M Jun 17 '15 at 8:43
  • $\begingroup$ Not around that time, invariant approach began with Cartan in 1920s, but invariant definition in terms of multilinear forms on Cartesian products of copies of a space and its duals is equivalent to the transformational definition, which refers to expressions of such forms in coordinates, see hsm.stackexchange.com/questions/599/… $\endgroup$ – Conifold Jun 17 '15 at 22:55
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    $\begingroup$ @Conifold Are you sure it wasn't Levi-Civita who motivated the subject via invariance of the product of multi-linear forms with their duals first? Certainly in the chapter I've taken the quote from, it makes clear from the start the idea of the invariance of energy to coordinate transformations and extends this to the product of multi-linear forms and their duals in general. Secondly, Einstein gives a brief overview of tensors in his 1915 paper, and defines covariant tensors via the invariance of their product with the corresponding contravariant tensor. $\endgroup$ – Larry Harson Jun 20 '15 at 23:00
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    $\begingroup$ The beginnings of $m$-fold covariance and contravariance go back at least as far as Christoffel, for example "Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades", 1869. The Levi-Civita book which you mention was the 1926 translation of his 1925 book, based on ideas in the famous Ricci and Levi-Civita article in 1900 on absolute differential calculus. The tensorial ideas were already clear in Bianchi's works, which rested on Christoffel's papers. $\endgroup$ – Alan U. Kennington Jun 28 '15 at 7:46

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