I was asked recently if the tensor product symbol $\otimes$ had been used before Bourbaki's publication on multilinear algebra in 1948 (a draft of this document can be seen at http://sites.mathdoc.fr/archives-bourbaki/PDF/040_iecnr_049.pdf starting on p. 10). I was unable to find any earlier usage, so that leads to my question:

Who introduced the symbol $\otimes$ for use in tensor products?

  • $\begingroup$ While we're at it, when was that exact same symbol used to indicate a basis vector pointing "down into the page" in vector diagrams? $\endgroup$ Feb 1, 2018 at 12:45

1 Answer 1



The earliest usage we have been able to find is in a 1936 paper by Francis J Murray and John von Neumann, entitled "Rings of Operators"(Annals of Mathematics, Vol. 37, pp. 116-229). Paul Halmos later used it in the earliest published version of his "Finite Dimensional Vector Spaces," published by Princeton University Press in 1942 (see Appendix II-Direct Products).

Jim Casey

  • $\begingroup$ Ok, so we have an earlier reference. Does that book indicate whether the symbol was invented by the author (or that he got it from somewhere else)? $\endgroup$ Feb 5, 2018 at 13:38
  • $\begingroup$ Keith and I located an earlier occurrence in a paper by Murray and von Neumann, "On Rings of Operators," published in 1936. $\endgroup$
    – Jim Casey
    Feb 6, 2018 at 14:32
  • $\begingroup$ Cool -- you should edit your answer to reflect that. $\endgroup$ Feb 6, 2018 at 16:44

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