This problem is a bit of a digression (at least it seems so) from the problems about imbeddings I'm dealing with currently (and I yet have a few more to complete).
Let X and Y be spaces. Define e : X x C(X, Y) --> Y with e(x, f) = f(x). e is called the evaluation map, and C(X, Y) is the set of all continuous functions from X to Y. If d is a metric for Y and if C(X, Y) has the corresponding uniform topology, then e is continuous.
The Attempt at a Solution
Now, if I defined a metric on X x C(X, Y), and if I considered the topology induced by this metric, the solution is quite easy, at least it seems so. But the problem is, I can't do that, right? And even if I'd do so (if the problem would be formulated that way), I don't see where the uniform topology on C(X, Y) jumps in.
So, any suggestions on how to attack this?