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Is there any condition for $X$ to be a compact space in the theorem "Space $C[X,R]$ is a closed subspace of $B[X,R]$? Because I see that if $X$ is not compact then $C[X,R]$ does not become a subset of $B[X,R]$. Counterexample is that $X=(0,1)$ is a metric space with usual metric and is not compact. If I define $f(x)=1/x$ on $X$, then it is continuous but not bounded.

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If you are defining $C[X,\mathbb R]$ as the space of all continuous functions on $X$ then this space is not a subset of $B[X,\mathbb R]$ in general. If $X$ is compact then $C[X,\mathbb R]$ is a subset of $B[X,\mathbb R]$. Assuming that $X$ is a metric space, then the converse is also true: if $C[X,\mathbb R]$ is a subset of $B[X,\mathbb R]$ then $X$ is compact.

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