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What is meant by being dense in spaces? Is that similar to the definition of density in topology?

Let $c_{00}$ be the space of all complex sequences with at most finitely many non-zero terms.

Show that the space $c_{00}$ is dense in $l_2$, but not dense in $l_\infty$.

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It’s exactly the topological notion of dense subset. It’s clear that as sets $C_{00}\subseteq\ell_2$ and $C_{00}\subseteq\ell_\infty$; you’re being asked to show that $C_{00}$ is a dense subset of $\ell_2$ in the $\|\cdot\|_2$ topology but not a dense subset of $\ell_\infty$ in the $\|\cdot\|_\infty$ topology.

HINTS:

  1. If $x=\langle x_k:k\in\Bbb N\rangle\in\ell_2$, for each $n\in\Bbb N$ let $y^{(n)}\in\ell_2$ be the sequence $$\langle x_0,x_1,\dots,x_n,0,0,\dots\rangle\;,$$ and show that $\lim_{n\to\infty}\|y^{(n)}-x\|_2=0$.

  2. Show that $\langle 1,1,1,\dots\rangle\in\ell_\infty$ has an open neighborhood disjoint from $C_{00}$.

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This means topologically. If $D$ is dense in $X$ then every neighborhood of a point in $X$ meets $D$. (meeets means intersects nonvoidly)

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