Consider the following situation: Let $X$ be a separable metric space [if this helps: I am mainly interested in the case $X = \mathcal{P}(\mathbb{R}^d)$, the space of all Borel-probability measures on $\mathbb{R}^d$ with the topology of weak convergence of measures], $T$ a positive number and consider a function $F: X^{[0,T]} \to \mathbb{R}$, which is sequentially continuous when considering the product-topology of weak convergence of measures on $X^{[0,T]}.$
Note: The space $X^{[0,T]}$ is not first-countable (correct?), hence it is not clear whether $F$ is continuous.
My question: Does $F$ take its supremum on a compact subset $A \subseteq X^{[0,T]}$?
I could not find anything in the literature on this. I tried to show that $F$ is upper semicontinuous, but couldn't do that and I do not have another idea either.
I would very much appreciate any help!
If $A$ is sequentially compact, then $F[A]$ is sequentially compact in $\mathbb{R}$ and thus compact (as the reals are a metric space in which these notions coincide) and $F[A]$ has a maximum, so assumes its supremum, say $a$. In the comments there was a further question on $A'=F^{-1}[\{a\}]$ which is easily seen to be sequentially closed when $F$ is sequentially continuous, and thus also sequentially compact as a sequentially closed subset of $A$.
It will be hard to apply this criterion, I think, as it's tricky to characterise sequentially compact subsets of a large product like $X^{[0,T]}$ which is itself non-sequentially compact (and not a sequential space either), e.g. many $\Sigma$-products are dense and sequentially compact in this product, as is well-known. I believe some work has been done on this for $C_p$-spaces but I'm not aware of a general characterisation beyond the definition.