I'm writing a piece with a lot of N-dimensional tensors $T \in \mathbb{R}^{n_1 \times n_2 \times \cdots}$. Most of the time the the actual shape, i.e. the values of $n_1$, $n_2$, and so on, are irrelevant and thus I don't want to carry them with me.
Consider for example the case that I want to state that every value of $T$ lies in the interval $[0, 1]$:
$T = [t_{i,j,\cdots}]_{n_1 \times n_2 \times \cdots}$ with $t_{i,j,\cdots} \in [0, 1]$
This is tedious if done multiple times without adding any information.
I'm looking for a notation that basically says "every element in this tensor".
Imagine this is denoted by a circle over the variable. The above could be denoted by:
$\overset{\circ}{T} \in [0, 1]$
- Is there a short (preferably as short as my) notation that does what I described above and that I'm unaware of?
- If not, have you suggestions how I could denote it? In the above example I used the circle as symbol, since it is commonly used within the hadamard products or the analogous operations, which are also applied elementwise.
Best Answer
You can use the notation for tensors with an undetermined number of indices
$$T_{\mu_1 \cdots \mu_p} \in [0,1]$$
where $\text{rank}(T)=p$