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How do I denote that I am aggregating values into a vector?

For example, I have a set of values $F$ in which a function $fun$ will be applied. Finally, a vector is build with the results.

For example, if $F = \{a,b\}$ and $fun(a) = 1, fun(b) = 2$, how can I denote that I'm assembling the vector: [1 2]?

Will a cartesian product suffice?

$$ V = \prod_{e \in F} fun(e) $$

Or should I use other operation?

f.leno
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  • Exactly what do you mean by "vector"? The space of maps $F\mapsto \mathbb R$ under pointwise addition and scalar multiplication is certainly a vector space in which $fun$ is a vector - and, even more, this tends to be a productive way to think about vectors when they arise this way. – Milo Brandt May 31 '16 at 04:41
  • I guess you mean tuple rather than vector. A tuple is a finite ordered list of elements, whereas a vector, which may be represented as a tuple, is a an element of a vector space. – Björn Friedrich May 31 '16 at 05:32

1 Answers1

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You may want to order the elements of $F$ into a given vector $\mathbf{f} \in F^n$, where $F^n = F \times F \times ... \times F$ $n$ times (that is, if the elements of $F$ are distinct; otherwise you are using a multiset). This will allow for a reasonable input representation for a vector-valued function.

To indicate that you are assembling a vector, you may want to establish a vector-valued function $\mathbf{v} : F^n \rightarrow \mathbb{R}^n$ that is defined such that

$$\mathbf{v}_i(\mathbf{f}) = fun(\mathbf{f}_i), i \in [n].$$

This is a reasonable way to indicate that you are assembling a vector using the output of the function $\mathbf{v}$.

Steven
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