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It has been some time since I left university...

In a not too formal language, an $n$-dimensional vector is an indexed set of numbers $\{i_1, ..., i_n\}$. A $n\times m$ matrix is a set of numbers with a two-dimensional index $\{i_{11},...,i_{n1},i_{m1},...,i_{mn}\}$.

What is the generalization of this, i.e. a set of numbers with an $n$-dimensional index? What is it called?

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4 – Tom Cooney Jun 8 '12 at 11:35
Yes! That's it. If you'd given this as an answer, I'd chosen it. Thanks a lot! – Axel Jun 10 '12 at 19:30

Actually, you should think of a vector as a function $$x: \{1, 2, 3, .... , n\}\rightarrow\mathbb{R}.$$ The right abstraction is to think of a vector or a matrix as a function of this sort. A matrix is a function $$A: \{1, 2, 3, .... , m\}\times \{1, 2, 3, .... , n\}\rightarrow\mathbb{R}.$$

So an matrix of higher dimensions is just a function from a cartesian product of finite integer sequences to $\mathbb{R}$.

The usage of $\mathbb{R}$; is pro-forma. You can use any set of objects in its stead.

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When you say "think of a vector as a function", shouldn't that be a covector? At best I could see treating a vector as a function from $\{1,2,3,....n\}\rightarrow \mathbb{R}^n$, not to $\mathbb{R}$. – Robert Mastragostino Jun 8 '12 at 13:15
The answer that I've been looking for was given by Tom Cooney in the comment to my question. While the function analogy is alright, I think speaking of a tensor would be more common (I have some formulars that work for 2 and 3 dimensions, consisting of vectors for the 2-dimensional case, and of matrices for the 3-dimensional case. In n dimensions, it would be nth-order tensors.) – Axel Jun 10 '12 at 19:47

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