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  1. You start by defining a vector space at each point of the manifold. The defining feature being the vector transformation law under change of co-ordinates.

Then you define dual vectors as linear functions from vectors to reals. The "function aspect" of a dual vector becomes its defining feature. The transformation law becomes a derived feature.

Then you define tensors as linear functions from $k$ vectors and $l$ dual vectors to numbers. Again, the "function" aspect becomes the defining feature of the tensor. The tensor transformation law is a derived/implied feature.

All this treats vectors as the most fundamental, and dual vectors and tensors as derived/ defined in terms of vectors.

  1. You define vectors as (linear) functions from dual vectors to reals. And you define dual vectors as functions from vectors to reals. Tensors are defined same as in #1.

This treats vectors and dual vectors as equally fundamental. But these definitions seem circular as you're defining vectors in terms of dual vectors and dual vectors in terms of vectors.

  1. You define two independent vector spaces at each point of the manifold, one of vectors and one of dual vectors. The defining feature being the vector/dual vector transformation laws.

You then define the inner product operation between a vector and a dual vector (to get the "function aspect"). Tensors are then defined same as in #1.

All this treats vectors and dual vectors as equally fundamental. Tensors are derived objects here. The defining feature of vectors and dual vectors is taken to be the respective transformation laws.

The defining feature of a Tensor is taken to be its "function aspect". The transformation law of the tensor is a derived feature.

  1. You start by assigning an infinite number of independent vector spaces to each point of the manifold. The dimension of each vector space is required to be a power of $n$ : $n^r$, where $n$ is the dimension of the manifold. Multiple vector spaces of the same dimension are allowed.

A member of any of these vector spaces is defined to be a tensor. The defining feature being the tensor transformation law.

Tensors are treated as the most fundamental here. Vectors and dual vectors reduce to special cases of tensors.

The defining feature of every quantity becomes its transformation law. The "linear function" aspect of the tensors (like the inner product) has to be defined separately.

Which one is the most logical treatment?

Níckolas Alves
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Egg Man
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    This question seems likely to be answered in an opinion-based way, since there is a bunch of opinion going on in which of these approaches is "the most logical". In my opinion, none of them. To define anything in terms of transformation laws is to rely on coordinate systems, which are completely arbitrary. Instead, one can define vectors in a more abstract way (see Wikipedia) and then proceed as in your first approach. – Níckolas Alves Apr 11 '22 at 02:32
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    2 seems illogical. You define vectors using dual vectors, and dual vectors using vectors. – MBN Apr 11 '22 at 06:44
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    @MBN it is phrased in a weird manner, but it's perfectly fine if interpreted correctly. Basically one considers the ideal $I_x$ of all smooth functions $f:M\to\Bbb{R}$ such that $f(x)=0$, and considers the quotient vector space $I_x/I_x^2$, and defines $T_xM:=(I_x/I_x^2)^$. Then, $T_x^M=(I_x/I_x^2)^{**}\cong I_x/I_x^2$, so the original object considered, $I_x/I_x^2$ , is actually an equivalent way of defining the cotangent space. In this way, cotangent space is actually arises first, and the tangent space is defined as its dual, all the while invoking the double duality. – peek-a-boo Apr 11 '22 at 07:09
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    @peek-a-boo I agree with you ( https://physics.stackexchange.com/questions/105347/is-it-foolish-to-distinguish-between-covariant-and-contravariant-vectors/105780#105780 ) but I don't think that is what he meant. – MBN Apr 11 '22 at 08:21
  • @NíckolasAlves You mean we should define vectors as directional derivative operators operating on scalar functions on the manifold? – Egg Man Apr 12 '22 at 09:13
  • @EggMan precisely – Níckolas Alves Apr 12 '22 at 17:34

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