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For $\Omega$ an arbitrary set the family $C(\Omega, \mathbb K)$ of all functions $\Omega \to \mathbb K$ becomes a complete topological $\mathbb K$-algebra under the topology of uniform convergence. Note this convergence cannot be given by a norm because -- while $C(\Omega, \mathbb K)$ certainly contains the Banach space of bounded functions as a little island clustered around zero -- if you swim (infinitely) far enough from the shore you reach an unbounded function $f$, and clustered around it a second island of all functions $h$ for which $f-h$ is bounded. $C(\Omega, \mathbb K)$ is the archipelago of all such islands.

My question is, does $C(\Omega, \mathbb K)$ have any universal property? Is there a nicely axiomatised class of topological $\mathbb K$-algebras that are exactly the closed subalgebras of $C(\Omega, \mathbb K)$? I apologize if this is common-knowledge. I have been unable to find even a standard name for $C(\Omega, \mathbb K)$.

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  • $\begingroup$ If $\Omega$ is infinite, then $A=C(\Omega,\mathbb K)$ cannot be a topological algebra with the topology of uniform convergence since $t\,x\not\to 0$ in $A$ as $t\to 0$ in $\mathbb K$ if $x$ is unbounded. Only the addition is continuous making it an Abelian topological group. $\endgroup$
    – TaQ
    Oct 6, 2015 at 13:38

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