I would like to describe a real vector space with dimension $a \times b$ (as in $a$ times $b$). Is it correct to describe it with $$ \mathbb R^{a \times b},$$ or would that imply the space of matrices of dimension $a$ by $b$? What about $$ \mathbb R^{a \cdot b} \,?$$
The $\times$-sign reads nicer but I'm not entirely sure if it's used correctly. Thank you for your help!
Edit: To clarify, what I'm looking for is the case $$ \mathbb R^{3 \times 2} = \mathbb R^{6} $$
On
I have started having the opinion that mathematical notation is never really formal---unless you write it down using first-order logic, which is not something you should desire. As long as you're not doing axiomatic set theory, asking the question what mathematical objects really are is both unnecessary and has a really tedious answer.
Take $\mathbb R^n$ for example. What does that mean? In set theory, we often define $(x,y) = \{\{x\},\{x,y\}\}$ and $X\times Y = \{(x,y)\mid x\in X,y\in Y\}$. So $X^2 = X\times X$ by construction. But what about $X^3$? We may choose either $X\times(X\times X)$ or $(X\times X)\times X$, and for larger values of $n$, there are many different choices of definition of $X^n$. It is impossible to make a definition that allows us to have $X^m\times X^n = X^{m+n}$ (at least I have never seen one that does not include similar ambiguity).
From one confusing point to another: How do we define the set of $m\times n$ matrices over $\mathbb R$? We may define them as $(\mathbb R^m)^n$, but we run into the same problems as before. Alternatively, we can define them as the maps from $\{1,2,\ldots, m\}\times\{1,2,\ldots,n\}\to\mathbb R$, but then we still have to make identifications, namely of $\mathbb R^n$ with the the $n\times 1$-matrices.
How to solve this? To make life easier, we identify all of these definitions so that we do not have to worry about such problems. This "identification" is a concept that is impossible to formalise, since it merely means that they are in fact different, but that we shall nevertheless write an equality sign between them and use the same notation to refer to both of them. If we were to be so pedantic as to not allow this, dealing with mathematics would have far more tedious definitions and far less actual, interesting content. Hence from now on, $X^n\times X^m = X^{m+n}$, and the matrices may be defined as sets or mappings or whatever as long as we know what we are talking about.
Conclusion: Every notation is perfectly acceptable, the only problem being that it might end up confusing people. So always explain your notation.
Start by noting that given $a,b\in \mathbb N$, there are vector spaces with dimension $a\times b$ different from $\mathbb R^{a\times b}$. So by restricting yourself to $\mathbb R$, you're already committing some abuse of notation.
As for the symbol $\mathbb R^{a\times b}$ it denotes $\{(x_1, \ldots ,x_{a\times b})\colon x_1, \ldots ,x_{a\times b}\in \mathbb R\}$. It is not a set of matrices (not in the sense that you mean).
Regarding $\mathbb R^{a\cdot b}$, it's the same as the above, you're just using $\cdot$ instead of $\times$ to denote the usual multiplication.
Having said all this, it "doesn't matter" if you choose $\mathbb R^{a\times b}, \mathbb R^{a\cdot b}, \mathcal M _{a\times b}(\mathbb R)$ or any other vector space with dimension $a\times b$ over $\mathbb R$ because any two vector spaces over a field $\mathbb F$ having the same dimension are isomorphic.