Basic rough path definitions and calculations/references I tried reading Friz and Hairer's book called A course on Rough Paths. Unfortunately, I am not really familiar with tensor product spaces and hence confused about some of the notation/procedures/definitions. I tried finding them on this site and on Wikipedia, but couldnt.
Looking at Exercise 2.1, the following quantity is defined:
$$ \mathbb{X}_{s,t} := \int _ s ^t X_{s,r} \otimes \dot{X}_r dr $$, where X  is a smooth valued path with values in $V$. I am not really sure how to work with that quantity, since the integrand is some object in $V \otimes V$.
Looking at other notes (https://pub.ist.ac.at/~gzanco/A_brief_introduction_to_rough_paths.pdf , page 7), I would imagine that this quantity is equal to $\int _ s ^t X_{s,r} \otimes d X_r$, (also referencing to (2.2) in the book) which I cannot give rigorous meaning to, either. From the context of everything that I've read I would assume that if $V$ is of dimension $d$, then the integral is somehow defined as the cartesian product of the coordinates written in a matrix, e.g.
$$ \left( \int _ s ^t X_{s,r} \otimes d X_r \right)_{i,j} = \int _ s ^t X_{s,r}^i  d X_r ^j  , \: \: \: \: \: \:  i,j \leq d.$$
This would make sense, since $V \otimes V \simeq R^{dxd}$ if $V$ is $d$-dimensional and $\mathbb{X}: [0,T]^2 \rightarrow V \otimes V$ is then true & the proof works with this definition. Is this correct? Does that mean that I can work with $\int _ s ^t X_{s,r} \otimes d X_r $ in a local fashion, e.g. look at the coordinate integrals in general (as soon as the integrals make sense)? At last, where do I find such definitions? I couldnt find them in the aforementioned book, neither in a monograph by one of the authors. I feel like I have a huge knowledge gap here and would like to know how I can fix that.
I would appreciate any clarification on the matter/resources that I can look at and understand these objects better.
 A: You are essentially correct in what you write about coordinates. Before I explain, let me just say that the tensor notation is extremely useful in writing up everything in a concise and tidy fashion. Moreover, the tensor algebra used is an example of a relevant algebraic structure which comes into play when one talks about generalisations of rough paths (such as branched rough paths a la Gubinelli).
But lets get back to your question.
I will from now on always assume that $V = \mathbb{R}^d$ is finite dimensional with $e_i , i \in \{1,\ldots,d\}$ being the standard basis (or any basis it does not matter). Moreover, we do some calculations for a smooth path $X \colon [0,T] \rightarrow V$ and we write $X_t = \sum_{1\leq i\leq d}X^i_te_i$ and $X_{s,t} = X_t-X_s$ for the increment (superscripts for coordinates, subscripts for evaluation at time $t$).
Sidenote: All of this works for Banach spaces, but then you do not have coordinates to break it down and you need to work with (topological) tensors.
Lets assume first that $d=1$. Then we have
$$\int_s^t X_{s,r} \otimes dX_r = \int_s^t X_{s,r} \dot{X}_r \mathrm{d}r $$
where multiplication makes sense now in the reals and we can take the derivative of $X$ as $X$ is smooth (enough). The integral on the right hand side also makes sense in the Riemann-Stieltjes or Young sense due to our regularity assumptions.
Now in $1$d everything is simple (also you might note that we have actually used the natural identification $\mathbb{R}\otimes \mathbb{R} = \mathbb{R}$ (tensor product over the reals).
If $d>1$ we have to give meaning to the integral. Obviously we have a bunch of coordinate functions (the $X^i$ for $1\leq i \leq d$ here) which we can integrate against each other as in the onedimensional case. The idea is now that the tensors allow us to conveniently keep track of what was multiplied against what. Since $e_1,\ldots, e_d$ is a basis of $\mathbb{R}^d$, $e_i \otimes e_j, 1\leq i,j\leq d$ is a basis of $\mathbb{R}^d \otimes \mathbb{R}^d = \mathbb{R}^{d^2}$.
So the meaning of the integral
$$\int_s^t X_{s,r} \otimes dX_r = \sum_{1\leq i,j \leq d} \int_s^t X^i_{s,r} \dot{X}^j_r \mathrm{d}r e_i \otimes e_j$$
Note that we have only used the tensors here to organise and remember what we integrated against what. If we again identify $\mathbb{R}^d \otimes \mathbb{R}^d$ with the space of $d \times d$ matrices you recover exactly your presentation of the integral as a matrix valued integral. Of course the idea is now to consider also iterated integrals (whence we get triple, quadruple,... tensorproducts) and these are more complicated then matrices. In the end it all boils down at the level you are struggling with to two things:

*

*Integration (in the Young-Sense) of paths against each other (here we invest heavily smoothness)

*bookkeeping with tensors (just remember the indices)

I think this is maybe easier to grasp from the old book by Friz and Victoire who compute in coordinates. Alternatively you could take a look at Chapter 8 of my book project where this is also explained in detail (though I think I was also a bit sloppy in explaining the tensor notation).
