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I'm trying to understand the set [itex]C_0(X)[/itex], defined here as the set of continuous functions [itex]f:X\rightarrow\mathbb C[/itex] such that for each [itex]\varepsilon>0[/itex], [itex]\{x\in X|\,|f(x)|\geq\varepsilon\}[/itex] is compact. (If you're having trouble viewing page 65, try replacing the .se in the URL with your country domain). The book also defines [itex]C_b(X)[/itex] as the set of all bounded continuous functions from X into [itex]\mathbb C[/itex]. The book makes additional assumptions about X, but clearly the definitions work even when X is just an arbitrary topological space.
It's always hard work to fill in the details that Conway leaves out. I think I have verified that [itex]C_0(X)\subset C_b(X)[/itex], and that if X is Hausdorff, [itex]C_0(X)[/itex] is closed under linear combinations. I still need to show that [itex]C_0(X)[/itex] is a closed set, and I would like to understand what any of this have to do with local compactness. [strike]Is [itex]C_0(X)[/itex] a subalgebra or just a vector subspace?[/strike]. I would appreciate any help with any of these details.
I have already LaTeXed the proof of the first part ([itex]C_0(X)[/itex] is a linear subspace, if X is Hausdorff) for my notes. I'll post it here if someone requests it. I have also LaTeXed the proof that [itex]C_b(X)[/itex] is a Banach algebra with identity, and wouldn't mind posting that too.
Edit: I think I proved that [itex]C_0(X)[/itex] is closed under multiplication (if X is Hausdorff), so we can scratch that item off the list.
It's always hard work to fill in the details that Conway leaves out. I think I have verified that [itex]C_0(X)\subset C_b(X)[/itex], and that if X is Hausdorff, [itex]C_0(X)[/itex] is closed under linear combinations. I still need to show that [itex]C_0(X)[/itex] is a closed set, and I would like to understand what any of this have to do with local compactness. [strike]Is [itex]C_0(X)[/itex] a subalgebra or just a vector subspace?[/strike]. I would appreciate any help with any of these details.
I have already LaTeXed the proof of the first part ([itex]C_0(X)[/itex] is a linear subspace, if X is Hausdorff) for my notes. I'll post it here if someone requests it. I have also LaTeXed the proof that [itex]C_b(X)[/itex] is a Banach algebra with identity, and wouldn't mind posting that too.
Edit: I think I proved that [itex]C_0(X)[/itex] is closed under multiplication (if X is Hausdorff), so we can scratch that item off the list.
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