Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 3 / c (The coordinate vectors in finitely supported sequences)

Take any finite selection of distinct indices $i_1,\dots ,i_r$ and a relationThe coordinate at $i_j$ on the left is exactly ${\alpha }_j$, while the zero vector has coordinate zero. Hence ${\alpha }_j=0$ for each $j$. Every finite subset of $E$ is therefore linearly independent, which is the definition of linear independence for an infinite set. Thus $E$ is linearly independent in the vector space of finitely supported sequences.

Every $v\in V$ is a finitely supported sequence, so for a finite set of indices $S$,Thus $E$ is a spanning set. If $F$ contained $v\notin E$, the displayed expression would give a nontrivial relationamong the distinct vectors of the finite subset $\{v\}\cup \{e^{(i)}:i\in S\}$ of $F$: the coefficient of $v$ is $1$. This contradicts the linear independence of $F$. It also covers $v=0$, whose singleton is not linearly independent. Hence $F=E$. In particular, $E$ is both a basis and a maximal linearly independent subset of $V$.