I'm trying to fulfill the details in a proof whose aim is to guarantee the existence of a compact space with certain properties, but in this case I had no success.
Let $X$ be a metric space (actually, the author requires completely regular and Hausdorff, but I'm not sure how that could work) and $G$ a locally compact second countable group with Haar measure $\lambda$ (I state it, though I suspect it is not crucial in what follows).
Let $B(X)$ denote the Banach space of measurable bounded real-valued functions on $X$ with norm $\lVert f\rVert=\sup_{x\in X}\lvert f(x)\rvert$ and $L^1(G)$ the Banach space consisting of $\lambda$-summable functions equipped with the norm $\lVert \varphi \rVert_1=\int_G \lvert\varphi(g)\rvert\,d\lambda(g)$.
For any $\emptyset\ne D\subseteq B(X)$, define $Y(D)$ to be the set of all real-valued functions $\gamma$ on $L^1(G)\times D$ satisfying the following conditions: (i) for each $f\in D$, $f\mapsto \gamma(\varphi,f)$ is linear wrt $\varphi\in L^1(G)$; (ii) $\lvert\gamma(\varphi,f)\vert\le \lVert\varphi\rVert_1\cdot\lVert f\rVert$. It is clear that $Y(D)$ is nonempty.
Question 1: Show that, equipped with the topology of pointwise convergence, $Y(D)$ is compact Hausdorff.
It is clear that $Y(D)$ is Hausdorff. Concerning the compactness, since the second countability is not an hereditary property, I have tried to show that every net has a cluster point (this seems to me the easiest way, isn't it?). I'm not very comfortable with nets (we never introduced them in our topology course). Anyway, here is my attempt. Since the pointwise convergence is given, we have that if $(N,\le)$ is a directed set, $n\in N$ and $\gamma_n,\gamma\in \Bbb{R}^{L^1(G)\times D}$, the following holds: $$\gamma_n \to \gamma \Leftrightarrow \gamma_n(\varphi,f)\to\gamma(\varphi,f)\,\forall (\varphi, f)\in L^1\times D.$$ If $\gamma_n\in Y(D)$, by assumption $\gamma_n(\varphi, f)$ is bounded in $\Bbb{R}$, so Bolzano-Weiestrass for nets implies the existence of a convergent subnet $\gamma_{n(k)}$ with limit a real number, say $\gamma(\varphi,f)$. (Here the directed set $K$ defining the subnet depends also on $\varphi$ and $f$, so it doesn't seem obvious how to get a subnet which guarantees that $\gamma_{n(k)}\to \gamma$. However, if I can do this, then) By using properties of limits in $\Bbb{R}$ I can show that $\gamma$ belongs to $Y(D)$.
As you can understand, I was not able to formalize in a precise way the argument, so can you help me to do this? I hope someone can help me to understand how to formalize this line of reasoning, whenever it is correct.
Question 2: After this, it is also stated that $D$ separable implies $Y(D)$ compact metrizable. I know that a compact space is metrizable iff it is Hausdorff and second countable, so I have to show that $D$ separable implies $Y(D)$ second countable. It should be a consequence of $\lvert\gamma(\varphi,f)\rvert\le \lVert\varphi\rVert_1\lVert f\rVert$, but I don't see how.
Edit: Reference: Varadarajan "Groups of Automorphisms of Borel spaces" (1963), p. 198