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The following discussion is for finite-dimensional vector spaces only:

The tenor product of two vector spaces $V$ and $W$ arises, because you want to work with bilinear, rather than just linear, maps and functions of $V \times W$. The key observation is that the space of all possible bilinear scalar-valued functions or vector-valued maps is itself a vector space that is spanned by simple bilinear functions constructed by multiplying a linear function on $V$ by a linear function on $W$.

Let $T$ denote the vector space of all scalar-valued bilinear functions on $V \times W$ and observe that there is a natural map $V^* \times W^* \rightarrow T$. Moreover, it is easy to see that $V^* \times W^*$ generates all of $T$ in the sense that its image does not lie in any proper subspace of $T$. So this leads to the notation $T = V^* \otimes W^*$

The next observation is that there is a natural bilinear map $V\times W \rightarrow T^*$ that corresponds to evaluation of the given $(v,w)$ with an element of $T$. Again, the image "spans" all of $T^*$ in the sense that it does not lie in any proper subspace. Moreover, the linear duality between $T$ and $T^*$ corresponds exactly to the evaluation of a bilinear function on an element in $V\times W$. So it is reasonable to denote $T^*$ by $V \otimes W$.

So, morally speaking, $V\otimes W$ is the "smallest" vector space such that there is a bilinear injective map $V \times W \rightarrow V\otimes W$. Of course, this statement can be made precise by defining $V\otimes W$ as a universal object.

Finally, it is not difficult to prove the rather cool (and for me not so obvious) observation that the space of linear maps $V \rightarrow W$ is naturally isomorphic to $W\otimes V^*$. In this sense, tensors generalize the matrices viewed as linear transformations.

The following discussion is for finite-dimensional vector spaces only:

The tenor product of two vector spaces $V$ and $W$ arises, because you want to work with bilinear, rather than just linear, maps and functions of $V \times W$. The key observation is that the space of all possible bilinear scalar-valued functions or vector-valued maps is itself a vector space that is spanned by simple bilinear functions constructed by multiplying a linear function on $V$ by a linear function on $W$.

Let $T$ denote the vector space of all scalar-valued bilinear functions on $V \times W$ and observe that there is a natural map $V^* \times W^* \rightarrow T$. Moreover, it is easy to see that $V^* \times W^*$ generates all of $T$ in the sense that its image does not lie in any proper subspace of $T$. So this leads to the notation $T = V^* \otimes W^*$

The next observation is that there is a natural bilinear map $V\times W \rightarrow T^*$ that corresponds to evaluation of the given $(v,w)$ with an element of $T$. Again, the image "spans" all of $T^*$ in the sense that it does not lie in any proper subspace. Moreover, the linear duality between $T$ and $T^*$ corresponds exactly to the evaluation of a bilinear function on an element in $V\times W$. So it is reasonable to denote $T^*$ by $V \otimes W$.

So, morally speaking, $V\otimes W$ is the "smallest" vector space such that there is a bilinear injective map $V \times W \rightarrow V\otimes W$. Of course, this statement can be made precise by defining $V\otimes W$ as a universal object.