Is the completion of $\{x=(x_n)|x_n\in \mathbb R \text{ and for a given } x,\text{ only finitely many } x_n\neq0\}$ equipped with the norm $\|x\|:= |x_1|+|x_2|+...$ simply the set of all real sequences?
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The "larger" space here is the set of all sequences $x$ such that $$\tag{1}\Vert x\Vert= \sum\limits_{n=1}^\infty |x(n)|$$ is finite. This is the space of absolutely summable sequences, denoted by $\ell_1$, with norm defined as in (1). One can show that $\ell_1$ is a complete normed linear space. Let $F$ be your set of sequences. Then $F$ is the space of all sequences of finite support and sits inside $\ell_1$. Moreover, given an element $x$ in $\ell_1$, the sequence $\{y_n\}$ in $F$ with terms defined by $$y_n=(x(1),x(2),\ldots, x(n),0,0,\ldots)$$ converges in norm to $x$. This shows that $F$ is a dense subset of $\ell_1$. As such, the completion of $F$ is $\ell_1$. |
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