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When defining tensors as multilinear maps, I am having trouble understanding why a tensor, let's say of type (2,1), can be written in the following way:

$$T = T^{\mu\nu}_{\rho} e_\mu \otimes e_\nu \otimes f^\rho$$

where $\{e_\mu\}$ is a basis for the vector space, and $\{f^\mu\}$ is its dual basis.

I know how to expand the tensor when applied to some generic vectors and covectors. So in the example above:

$$T(\alpha,\beta,X) = T(\alpha_\mu f^\mu,\beta_\nu f^\nu, X^\rho e_\rho) = \alpha_\mu \beta_\nu X^\rho T(f^\mu,f^\nu,e_\rho) = \alpha_\mu \beta_\nu X^\rho\, T^{\mu\nu}_{\rho}$$

How would I proceed from here? Can I just say $\alpha_\mu = \alpha_\nu f^\nu(e_\mu) = \alpha (e_\mu) $, and similarly for the others? Then I would get:

$$T(\alpha,\beta,X) = T^{\mu\nu}_{\rho}\, \alpha(e_\mu)\, \beta(e_\nu)\, f^\rho(X)$$

I am not sure how to generalise from here. Apologies if this is really obvious.

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You have the right idea--by the time you're at your last step, everything is just numbers. At this point, just einstein sum your indices.

I will say that most working people, unless they are deriving general results, will start from the tensor in an explicit coordinate basis, and infer the mathematical object. This way is more formally "correct", but a lot less useful for actually figuring out physics, exactly due to all of these gymnastics.

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