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I noticed that the rank of a tensor is a kind of "two-dimensional" property - the covariant components come first, then the contravariant. If I understand correctly, these components refer to the normal vector space and its dual counterpart. What is it, and why does vector space have some kind of dual counterpart, in simple terms? (Also, why is the norm of a vector space and its dual related by the formula 1/a+1/b=1?)

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    If there were "simple terms" then this wouldn't be a deep concept. When John Tate won the Abel prize, the local paper interviewed him. The reporter asked him to explain his work in layman's terms. John replied that he couldn't. It took years of concentration and training to master his field, did the reporter think that he could reduce it to a paragraph of easily digestible prose? Instead of making the concept easier, make your brain stronger. – B. Goddard Jun 11 '23 at 12:47
  • I'll dare to use a physical metaphor, which is perhaps not rigorous but more visual. Vectors can be seen as particles and linear forms of the dual space as antiparticles; when they meet, their annihilation produces a scalar, so that they can be thought as inverses (in the same way as integration and differentiation killing each other). – Abezhiko Jun 11 '23 at 13:19
  • You might find this post to be helpful. In a sense, you can always think of a tensor as being a multilinear map. Producing a vector as an output is, in some sense, equivalent to taking a dual-vector as an input. – Ben Grossmann Jun 11 '23 at 17:39
  • One way to think about the importance of dual vectors is that they allow us to treat an output as if it were an input – Ben Grossmann Jun 11 '23 at 17:49

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