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Frechet -> Fréchet
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LSpice
LSpice
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I believe that the following fact is true and I am looking for a reference.

Let $X$ be a locally compact Hausdorff topological space (may be assumed to be metrizable). Let $V,W$$V$, $W$ be Frechet Fréchet spaces. Then any separately continuous map $X\times V\to W$ which is linear with respect to the second variable, is (jointly) continuous.

I believe that the following fact is true and I am looking for a reference.

Let $X$ be a locally compact Hausdorff topological space (may be assumed to be metrizable). Let $V,W$ be Frechet spaces. Then any separately continuous map $X\times V\to W$ which is linear with respect to the second variable, is (jointly) continuous.

I believe that the following fact is true and I am looking for a reference.

Let $X$ be a locally compact Hausdorff topological space (may be assumed to be metrizable). Let $V$, $W$ be Fréchet spaces. Then any separately continuous map $X\times V\to W$ which is linear with respect to the second variable, is (jointly) continuous.

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asv
asv
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Separate continuity implies (joint) continuity

I believe that the following fact is true and I am looking for a reference.

Let $X$ be a locally compact Hausdorff topological space (may be assumed to be metrizable). Let $V,W$ be Frechet spaces. Then any separately continuous map $X\times V\to W$ which is linear with respect to the second variable, is (jointly) continuous.