I can't see how a processes $X=(X_{t})_{t\in T}$ is an element of $\ell^{\infty}(T)$ which is what Van der Vaart writes in the beginning of chapter $18$ in his book on asymptotic statistics. Given that we have observed some $\omega$, sure, then a path is in this space but if not then I don't see how this is something which you would say. I quote
"We are particularly interested in developing the theory(of convergence in distribution) for random functions, or stochastic processes, viewed as elements of the metric space of all bounded functions"
He goes on to keep this perspective a few pages ahead
"Thus, we may concentrate on weak convergence in the space $\ell^{\infty}(T)$"
This leaves me confused, could anyone explain why he formulate himself in this way?
A stochastic process $X$ is a function of two variables $(\omega,t)\mapsto X(\omega,t).$
One way to look at $X$ is that every $t$ gives a function $\omega\mapsto X(\omega,t)$, i.e., a stochastic process is a family of random variables.
Another way to look at $X$ is that every $\omega$ gives a function $t\mapsto X(\omega,t)$, i.e., a stochastic process is a random function on $T$. If $X$ is bounded, then this function belongs to $\ell^\infty(T)$ for every $\omega\in\Omega$.
Thus, $X$ is not an element of $\ell^\infty(T)$, it is a mapping from $\Omega$ to $\ell^\infty(T)$. The distribution of $X$ is a probability measure on $\ell^\infty(T)$, and convergence of such random variables corresponds to convergence of probability measures on $\ell^\infty(T)$.