TITLE: Tensors defined by transformation laws are tensors at a vector space or tensor fields?
QUESTION [2 upvotes]: In Physics it is common to see tensors defined by transformation properties relating components of the object in different coordinate systems.
There is, however, two ways we can think of a tensor: a tensor at a particular vector space (in a geometrical context, considering smooth manifolds, this would be a single tensor located at a point) and a tensor field (in a geometrical context, considering smooth manifolds, this would be a tensor located at each point).
The first point of view is: we have a vector space $V$, in that case a $(r,s)$-tensor is a multilinear map
$$T:V\times \cdots \times V\times V^\ast \times\cdots \times V^\ast \to \mathbb{R}$$
where there are $r$ copies of $V$ and $s$ copies of $V^\ast$.
The second point of view is: we have a smooth manifold $M$ with tangent bundle $TM$ and cotangent bundle $T^\ast M$. If $\Gamma(TM)$ is the space of sections of $TM$ and similarly $\Gamma(T^\ast M)$ is the space of sections of $T^\ast M$ a tensor field of type $(r,s)$ is a $C^\infty(M)$-multilinear map
$$T : \Gamma(TM)\times \cdots \Gamma(TM)\times \Gamma(T^\ast M)\times\cdots \times \Gamma(T^\ast M)\to C^\infty(M)$$
that is, it takes $r$ vector fields, $s$ covector fields and outputs a function, such that if $f\in C^\infty(M)$ we have
$$T(X_1,\dots, fX_i,\dots,X_r,\omega_1,\dots,\omega_s)=fT(X_1,\dots,X_r,\omega_1,\dots,\omega_s)$$
for any $i$ and similarly for the $\omega$ entries.
The question here is: the Physicists' traditional definition of tensors, seem in many mathematical physics textbooks, electrodynamics textbooks and many relativity textbooks, based on transformation properties, defines a tensor at a particular vector space or a tensor field?
I ask that because it is common to see the abuse of language of calling a tensor field just "tensor" and a vector field just "vector". I just want to know here whether that definition is meant to define a tensor or a tensor field.

REPLY [1 votes]: The "definition by transformation law" works for both tensors and tensor fields. A tensor is an element of $V\otimes\dots\otimes V$, a tensor field a section of $TM\otimes \dots\otimes TM$ (omitting the possibility of duals because they add no insight here). If you define a tensor by how it transforms under $\mathrm{GL}(V)$, then you defined a tensor in the mathematical sense, if you use $\mathrm{GL}(TM)$ (or, slightly less general, coordinate transformations of $M$ itself that induce transformations of $TM$), then you have defined a tensor field in the mathematical sense.
Of course, in the first case your "tensor" is constant while in the second case it's a function on $M$ - this dependence is often suppressed in the notation.
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