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I'd like to represent mathematically tuples having $n$ elements (where $n$ is not a constant) as vectorial space.

$$\mathbb{F} \times \underbrace{\mathbb{F} \times \mathbb{F} \cdots \mathbb{F}}_{\text{0 or more times}}$$

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Your answer is interesting, but keep in mind this is not a vector space. You could easily define $A+B$ by appending infinitely many zeros to both tuples and then doing the addition term by term. But even then, you must be cautious with $[0] = [0,0] = [0,0,0] = \dots$ so taking the union is not enough.

Your tuples may be considered as coefficients of polynoms, and you can work on $\Bbb F[X]$, which is the usual notations for polynoms in variable $X$ with coefficients in $\Bbb F$ (which should be a ring or a field). Of course you get more than a vector space, since $\Bbb F[X]$ is also a ring, but it's probably the closest to what you want.

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This is commonly written as $\mathbb F^{<\omega}$ or $\mathbb F^{<\mathbb N}$. It is the set of all finite tuples.

Generally speaking if $\kappa$ is a cardinal then $\mathbb F^{<\kappa}$ is the set of all sequences of length $<\kappa$.

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Found the answer while trying to explain what I want.

$$\bigcup_{i = 1 \cdots \infty} \mathbb{F}^i$$

Still if you known something more elegant any other answer is welcome

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