I have been thinking about vectors, vector spaces, tensors, and tensor spaces. So far, I have surmised that a tensor space is defined to be $(\otimes_{i=1}^{k} V,+)$ where the addition is vector addition relative to the component separation caused by the basis vectors and the scalar multiplication is over $\mathbb{R}$ or $\mathbb{C}$.
However, I'm confused. Are tensor spaces vector spaces? How do I visualize the notion of a tensor space? I can easily visualize a vector space with examples such as $\mathbb{R}^3$, but I am failing to understand how to visualize higher dimensional structures. Would someone please provide for me some examples of tensor spaces?
Hint:
Take $k=2$.
For an element $T$ in, say $\mathbb R^3\otimes\mathbb R^3$, let us illustrate.
The space $\mathbb R^3\otimes\mathbb R^3$ is generated by the 9 elements $$e_1\otimes e_1,\qquad e_1\otimes e_2,\qquad e_1\otimes e_3,$$ $$e_2\otimes e_1,\qquad e_2\otimes e_2,\qquad e_2\otimes e_3,$$ $$e_3\otimes e_1,\qquad e_3\otimes e_2,\qquad e_3\otimes e_3.$$
Then $T$ is a linear combination on these.
Symbolically is written as $T=T^{sr}\ e_s\!\otimes\!e_r$, that is a bi-indexed sum. Unfold is $$T=T^{11}\ e_1\!\otimes\!e_1+T^{12}\ e_1\!\otimes\!e_2+\cdots +T^{32}\ e_3\!\otimes\!e_2+T^{33}\ e_3\!\otimes\!e_3.$$
If $T=T^{sr}\ e_s\!\otimes\!e_r$ and $U=U^{sr}\ e_s\!\otimes\!e_r$ are two of them then $$T+U=T=(T^{sr}+U^{sr})\ e_s\!\otimes\!e_r,$$ is a vector sum, and $$qT=(qT^{sr})\ e_s\!\otimes\!e_r$$ is the scalar action.
Now you have a start to begin to grasp. Think with $k=3$ next.