Tensors defined by transformation laws are tensors at a vector space or tensor fields?

Question

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.

Answer 1

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.

Answer 2

just want to know here whether that definition is meant to define a tensor or a tensor field.

It's context dependent, in my limited experience. For example, you could check a single tensor is a tensor by applying transformations to it, in GR, SR, or QFT. But it is also used to describe a field if the the context is obvious ( to the author at least :)