# Why is the space of continuous mappings between two G-spaces a G-space

Let XXX and YYY be GGG-spaces for a group G (you may assume Hausdorff). Let YXYXY^X be the space of continuous mappings from XXX to YYY with the compact open topology. This space carries a conjugation action of GG.

It is often stated that under some minor regularity assumptions (X locally compact ?), this becomes a G-space, i.e. the action becomes continuous, but I have never seen a proof of this basic fact anywhere. I’d appreciate any hints!

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If the spaces G,X,YG,X,Y are “nice”, then you have a homoemorphism
Hom(G×YX,YX)≅Hom(G×YX×X,Y). \text{Hom}(G\times Y^X , Y^X )\cong \text{Hom}(G\times Y^X \times X, Y) .
So if you want to show that some map is contained in the set on the left side, you can also show that the map is contained in the right side.

Now what is the map you are looking for corresponding to? It should be a compistion of product maps of the GG-action on XX the evaluation and the inverse GG-action on YY.

So you need to ensure that the evaluation is continuous and the inverse map of GG is also continuous.

Wow, that does it! Thanks a ton! I guess I underestimated the power of adjunctions once again…
– mathematician
2 days ago