The natural state space for a process indexed by an arbitrary set \(T\) is \(\ell^\infty(T)\). Its uniform norm matches simultaneous inference, but the space is nonseparable whenever \(T\) contains an uncountable uniformly separated subset. Tight limits therefore live on smaller separable sets of uniformly continuous paths.
The Banach space \(\ell^\infty(T)\)
Definition. \[ \ell^\infty(T) =\left\{z:T\to\mathbb R: \|z\|_T=\sup_{t\in T}|z(t)|\lt\infty\right\}. \] It is equipped with the norm \(\|\cdot\|_T\).
Completeness. The normed space \(\ell^\infty(T)\) is complete.
Proof. Let \(z_n\) be uniformly Cauchy. For each \(t\), the scalar sequence \(z_n(t)\) is Cauchy, so define \(z(t)=\lim_nz_n(t)\). Given \(\varepsilon>0\), choose \(N\) with \[ \sup_t|z_n(t)-z_m(t)|<\varepsilon \quad(n,m\ge N). \] Letting \(m\to\infty\) gives \(\|z_n-z\|_T\le\varepsilon\). Also \(z\) is bounded because \(z_N\) is bounded and \(\|z-z_N\|_T\lt\infty\). Hence \(z_n\to z\) in \(\ell^\infty(T)\).
Evaluation and supremum maps
Lipschitz functionals. For every \(t\in T\), coordinate evaluation \[ \pi_t(z)=z(t) \] has operator norm one. Moreover, \[ |\|z\|_T-\|w\|_T|\le\|z-w\|_T, \] \[ \left|\sup_tz(t)-\sup_tw(t)\right| \le\|z-w\|_T. \]
Proof. The coordinate bound \[ |z(t)-w(t)|\le\|z-w\|_T \] gives continuity and norm one. For the supremum, \[ z(t)\le w(t)+\|z-w\|_T \] for every \(t\); take suprema and repeat with \(z,w\) interchanged. Apply the same argument to absolute values for the norm.
Thus process convergence in \(\ell^\infty(T)\) immediately yields convergence of supremum statistics. Integral and argmax maps need their own continuity conditions.
Nonseparability
Nonseparability for uncountable \(T\). If \(T\) is uncountable, then \(\ell^\infty(T)\) is nonseparable.
Proof. For each subset \(A\subseteq T\), let \(1_A\in\ell^\infty(T)\). If \(A\ne B\), then \[ \|1_A-1_B\|_T=1. \] There are uncountably many such indicators. A separable metric space cannot contain an uncountable set whose pairwise distances are bounded below by one: disjoint open balls of radius \(1/3\) would each need a distinct point from a countable dense set.
Nonseparability explains why a coordinatewise measurable process need not be a Borel random element. A tight Borel probability measure on a metric space is concentrated on a separable subset, so any legitimate tight limit effectively occupies a smaller path space.
Uniformly continuous path spaces
Definition. For a semimetric \(\rho\) on \(T\), let \[ UC(T,\rho) =\left\{z\in\ell^\infty(T): \lim_{\delta\downarrow0} \sup_{\rho(s,t)<\delta} |z(s)-z(t)|=0\right\}. \]
Separability under total boundedness. If \((T,\rho)\) is totally bounded, then \(UC(T,\rho)\), with the uniform norm, is separable.
Proof outline. For each \(m\), choose a finite \(2^{-m}\)-net \(T_m\). Approximate a uniformly continuous function by its values on \(T_m\), quantized to rational multiples of \(2^{-m}\), and extend by a fixed nearest-net rule. The union over \(m\) of these finite rational descriptions is countable. Uniform continuity makes the approximations converge in sup norm.
Arzelà-Ascoli geometry
Arzelà-Ascoli criterion. Suppose \((T,\rho)\) is totally bounded. A set \(K\subseteq UC(T,\rho)\) is relatively compact in the uniform norm if and only if it is uniformly bounded and uniformly equicontinuous: \[ \sup_{z\in K}\|z\|_T\lt\infty \] and \[ \lim_{\delta\downarrow0} \sup_{z\in K}\sup_{\rho(s,t)<\delta} |z(s)-z(t)|=0. \]
Proof. For sufficiency, choose a finite \(\delta\)-net \(t_1,\ldots,t_N\) so the common oscillation is below \(\varepsilon\). Uniform boundedness makes the vectors \((z(t_1),\ldots,z(t_N))\) totally bounded in \(\mathbb R^N\); finitely many representative vectors, together with equicontinuity, form a finite \(3\varepsilon\)-net for \(K\) in sup norm. Completeness then makes the closure compact. Necessity follows because compact sets are bounded, and a finite \(\varepsilon\)-net of uniformly continuous functions has a common continuity modulus.
This theorem is the deterministic model of empirical-process tightness: control finitely many net coordinates and a common modulus of continuity.
Asymptotic tightness in \(\ell^\infty(T)\)
Equicontinuity criterion. Suppose a semimetric \(\rho\) makes \(T\) totally bounded. If \(Z_n(t_0)=O_{P^\star}(1)\) for some \(t_0\in T\) and \[ \lim_{\delta\downarrow0}\limsup_n P^\star\left\{ \sup_{\rho(s,t)<\delta} |Z_n(s)-Z_n(t)|>\varepsilon \right\}=0 \] for every \(\varepsilon>0\), then \(Z_n\) is asymptotically tight in \(\ell^\infty(T)\).
Proof outline. Choose successively smaller scales \(\delta_m\) so the probability that the oscillation exceeds prescribed \(\eta_m\downarrow0\) is summably small in the limsup. Together with a high-probability bound at \(t_0\), these constraints define a uniformly bounded, uniformly equicontinuous set \(K\). Arzelà-Ascoli makes its closure compact. Slight enlargements of \(K\) contain \(Z_n\) with arbitrarily high lower probability, which is asymptotic tightness.
Gaussian limits
A centered Gaussian process \(G=\{G(t):t\in T\}\) has intrinsic semimetric \[ \rho_G(s,t) =\{\mathbb E(G(s)-G(t))^2\}^{1/2}. \] If \(G\) is tight in \(\ell^\infty(T)\), then \((T,\rho_G)\) is totally bounded after identifying zero-distance points, and \(G\) has a version with bounded uniformly \(\rho_G\)-continuous sample paths. Entropy and majorizing-measure conditions provide usable sufficient criteria for this bounded continuity.