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Suppose I have a map $\Phi:\mathbb{R}\rightarrow C_b(\Omega)$ where the right hand side is equipped with the compact open topology $\tau_{co}$. How to show that such a function is continuous with respect to $\tau_{co}$. I want to show this by the seminorms $(p_K)$ generating the compact open topology: $$\left\{p_K(f):=\sup_{x\in K}{|f(x)|}: K\subseteq\Omega\ \text{compact}\right\}$$ and not by the mean of the well-known subbase of this topology. Whats about convergence of sequences, how can that be expressed by seminorms? Thanks.

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Well, $C_b(\Omega)$ has the initial topology with respect to those seminorms $p_K$, so simply show that the compositions $p_K \circ \Phi$ are continuous for all $K$. See the Wikipedia article on the initial topology 1 to see how this works (specifically look at the 'characteristic property').

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