I am having problems in understanding a criterion for relative compactness given in a book (see below for details if you are interested) on SPDEs. However, I think it just invokes a pretty general result on relative compactness in uniform spaces. We have the following setting: $S$ is a uniform space with uniform structure induced by a family $\{ d_{\alpha} ~|~\alpha \in A \}$ of pseudometrics, as is every uniform structure. We furthermore assume that the family has the following property (in the book a family with that property is called "filtering"): for any $\alpha_{1}, \alpha_{2} \in A$, there exists $\alpha \in A$ such that $$ d_{\alpha}(x,y) \geq \max \{ d_{\alpha_{1}}(x,y), d_{\alpha_{2}}(x,y) \} ~ \forall x,y \in S. $$ Using these pseudo-metrics, the author goes on to define a generalised Skorokhod space (see below), but I am not sure if this is really relevant to my question (so if the criterion I am confronted with holds only in this function space; I include the details below), leading to another set of pseudometrics which he calls $\{ \tilde{\delta}_{T,\alpha} ~|~ \alpha \in A \}$ and which inherits the "filtering" property from the original set. He then continues: "We know therefore [as the derived family has the filtering property, too] that a subset $K \subset \mathbb{D}([0,T];S)$ is relatively compact if and only if it has the two properties:
- (Quasi-compactness of $K$). For every $\alpha \in A$, $\varepsilon > 0$ there exists a finite covering of $K$ by "balls" $\{ \omega ~|~ \tilde{\delta}_{T,\alpha}(\omega, \omega_{1}) < \varepsilon \}$, $i = 1, \ldots, N$, $\omega_{1} \in \mathbb{D}([0,T];S)$.
- $K$ is included in a complete subspace of $\mathbb{D}([0,T];S)$."
Unfortunately, he does not add a reference for this conclusion and I could not find a criterion for relative compactness in such uniform spaces in any of the books I consulted.
Does anyone have a reference to a text where this criterion appears and that contains a proof?
Many thanks in advance!
For more context: The book mentioned is Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces, 1988, the quoted sentence is on p. 65. The pseudo-metrics that define the so-called $J$-topology (or Skorokhod topology) on the set $\mathbb{D}([0,T];S)$ of all cadlag functions from $[0,T]$ into $S$ are defined by \begin{equation} \begin{split} \tilde{\delta}_{T,\alpha}(\omega_{1}(\cdot), \omega_{2}(\cdot) ) := &\inf_{\lambda \in \Lambda_{T}} \big[ \sup_{t \in [0,T]} d_{\alpha} (\omega_{1}(t) , \omega_{2} \circ \lambda(t)) \\ &+ \sup_{t \in [0,T]} |t - \lambda(t)| + \sup_{s \neq t} |\log \frac{\lambda(t) - \lambda(s)}{t-s}| \big], \end{split} \end{equation} where $\Lambda_{T}$ is the set of increasing homeomorphisms of the interval $[0,T]$.
A typical example of a space with that uniform structure is a Hilbert space $(H, (\cdot,\cdot) )$ but endowed not with the strong topology but with the weak topology and whose uniform structure is defined by the following filtering set of seminorms: Let $A$ be the family of all finite subsets of $H$ and define $$ p_{\alpha}(h) := \sup_{h' \in \alpha} | (h', h) |, \quad \alpha \in A, h \in H. $$