Let $Q$ be a space of indices, and let $(V, |\cdot|)$ be a Banach space of values. Define the function space $X = C(Q,V)$, and equip it with the topology generated by seminorms $\|x\|_D := \sup_{d \in D} |x(d)|$, for any compact $D \subseteq Q$.

Under what conditions on $Q$ is $X$ a Fréchet space? Is it sufficient that $Q$ be separable & locally compact Hausdorff? Or, do we need the stronger condition that $Q$ be $\sigma$-compact?

Furthermore, under what conditions is $X$ a nuclear Fréchet space?



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