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The following is an example in my lecture notes:

"Let $X$ be a locally compact topological space (that is, a topological space in which every point has a compact neighborhood). Then $C_0(X)=\{f \in C_b(X)| \lim_{x \to \infty} f(x)=0 \}$ is a closed subspace of $C_b(X)$, the space of bounded continuous functions, and hence a Banach space. The notion of the limit of $f(x)$ as $x \to \infty $ used here is defined as follows: $\lim_{x\to \infty} f(x) = A$ if and only if for every $\varepsilon > 0$ there exists some compact set $K \subset X$ with $|f(x)−A| < \varepsilon$ for all $x \in X \setminus K$. If $X = \mathbb N$ (with the discrete topology), one often writes $c_0 = c_0(\mathbb N) \subset C_0(\mathbb N)$ for this subspace of $l^\infty(\mathbb N)$."

Since I assume that $c_0 = C_0$ here, at the very end of the last sentence, should it say "...writes $c_0 = c_0(\mathbb N) \subset C_{\color{red}{b}}(\mathbb N)$ for this subspace of $l^\infty(\mathbb N)$."? Probably.

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I guess $c_0$ is defined as the sequences which converge to $0$. In this case, to be coherent with the general definition $c_0:=C_0(\Bbb N)$ and it is both contained in $C(\Bbb N)$ and $C_b(\Bbb N)$ (but of course the latter is more accurate).

(note that we don't have to really deal with continuity since the topology is discrete, $C(\Bbb N)$ is such the set of sequences of real or complex numbers)

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