Compact set on functions space

Question

Let $(D[0,T], X\times X)$ the set of cadlag functions from $[0,T]$ to $X\times X$.

If I have a compact subset $K$ in $(D[0,T], X)$ and another compact subset $H$ in $(D[0,T], X)$, is $K\times H$ a compact subset of $(D[0,T], X\times X)$?

Thanks

Answer 1

Let $f_i \colon [0,T] \to X\times X$ be cadlag functions, such that $(f_i)$ is an universal net in $D([0,T], X^2)$, and $\pi_1 f_i\in K$, $\pi_2f_i \in H$, where $\pi_j \colon X^2 \to X$ denotes the $j$-th projection.

As $(\pi_jf_i)$, $j = 1,2$ are universal nets in $K$ resp. $H$, they are convergent, say $\pi_j f_i \to g_j$. Define $g = (g_1, g_2) \colon [0,T] \to X^2$, then $f_i \to g$. Hence $$ \{ f \in D([0,T], X^2) \mid \pi_1 \circ f \in H, \pi_2 \circ f \in K\} $$ is compact if $H$ and $K$ are.