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Equicontinuity

In mathematics, particularly real analysis and functional analysis, equicontinuity is a property of a family of functions that extends the concept of uniform continuity to collections of functions, ensuring a uniform bound on their variations across the entire family. Formally, given metric spaces (X, d_X) and (Y, d_Y), a family \mathcal{F} of functions f: X \to Y is equicontinuous if for every \epsilon > 0, there exists \delta > 0 (independent of the specific function in \mathcal{F}) such that whenever d_X(x, y) < \delta for x, y \in X, then d_Y(f(x), f(y)) < \epsilon for all f \in \mathcal{F}.^[1] The notion of equicontinuity was introduced independently by the Italian mathematicians Cesare Arzelà in his works from 1882–1883 and Giulio Ascoli in 1883–1884, as a key condition for studying the convergence and compactness of sequences of functions.^[2] It forms one of the two pillars—alongside pointwise boundedness—of the Arzelà–Ascoli theorem, which asserts that a subset of the Banach space C(K) of continuous real-valued functions on a compact metric space K, equipped with the supremum norm, is relatively compact if and only if it is pointwise bounded and equicontinuous.^[1] This characterization generalizes classical compactness results like Bolzano–Weierstrass to infinite-dimensional settings.^[2] Equicontinuity and the associated theorem have profound implications across analysis, enabling proofs of existence in ordinary differential equations via the Peano existence theorem, facilitating approximations in the Peter–Weyl theorem for representations on compact groups, and supporting the study of uniform convergence in series of functions and dynamical systems.^[2] In broader functional analysis, it underpins the compactness of embeddings between spaces like Lipschitz functions into continuous functions, aiding spectral theory and operator semigroups.^[3]

Fundamentals in Metric Spaces

Definition

In mathematical analysis, equicontinuity is a property of a family of functions that generalizes the notion of uniform continuity to collections of functions, ensuring that the functions vary in a controlled manner uniformly across the family. Let (X, d_X) and (Y, d_Y) be metric spaces. A family \mathcal{F} of functions f: X \to Y is equicontinuous at a point x \in X if for every \varepsilon > 0, there exists \delta > 0 such that for all f \in \mathcal{F} and all y \in X with d_X(x, y) < \delta, it holds that d_Y(f(x), f(y)) < \varepsilon. The family \mathcal{F} is equicontinuous (or pointwise equicontinuous) if this condition holds at every point x \in X.^[4] When the codomain Y = \mathbb{R} (or \mathbb{C}), the condition at a point x can be reformulated using the supremum norm over the family: for every \varepsilon > 0, there exists \delta > 0 such that \sup_{f \in \mathcal{F}} |f(x) - f(y)| < \varepsilon whenever d_X(x, y) < \delta. A stronger notion is uniform equicontinuity, where for every \varepsilon > 0, there exists \delta > 0 such that for all f \in \mathcal{F} and all x, y \in X with d_X(x, y) < \delta, d_Y(f(x), f(y)) < \varepsilon; this requires the modulus of continuity to be independent of both the point x and the function f.^[5] Equicontinuity applies specifically to families of functions, distinguishing it from uniform continuity, which is defined for individual functions. For a singleton family \{f\}, pointwise equicontinuity holds if and only if f is continuous on X, while uniform equicontinuity holds if and only if f is uniformly continuous on X; thus, any single continuous function generates a trivially equicontinuous family in the pointwise sense.^[4]

Basic Properties

A family of functions \mathcal{F} from a metric space X to a metric space Y is equicontinuous at a point x_0 \in X if for every \epsilon > 0, there exists \delta > 0 such that d_X(x, x_0) < \delta implies d_Y(f(x), f(x_0)) < \epsilon for all f \in \mathcal{F}. If \mathcal{F} is equicontinuous at x_0, then each individual function f \in \mathcal{F} is continuous at x_0.^[6] On a compact metric space K, a family \mathcal{F} \subset C(K, Y) of continuous functions is equicontinuous if and only if it is uniformly equicontinuous, meaning the \delta in the definition of equicontinuity can be chosen independently of the point in K. To see this, cover K with finitely many balls of radius \delta/3 where equicontinuity holds locally, and use compactness to adjust for overlaps via the triangle inequality.^[7]^[6] Equicontinuity is preserved under uniform limits. Specifically, if \{f_n\} is a sequence of functions that is equicontinuous and f_n \to f uniformly on a set E, then the family \{f_n\} \cup \{f\} is equicontinuous on E. The proof follows by passing the \delta from the equicontinuity of \{f_n\} to the limit function, using the uniform convergence to control the difference |f(x) - f(y)| \leq |f(x) - f_n(x)| + |f_n(x) - f_n(y)| + |f_n(y) - f(y)|.^[7] If \mathcal{F} and \mathcal{G} are uniformly equicontinuous families, with \mathcal{F} from X to Y and \mathcal{G} from Y to Z, then the family of compositions \{g \circ f \mid f \in \mathcal{F}, g \in \mathcal{G}\} is uniformly equicontinuous from X to Z. For \epsilon > 0, choose \delta_Y for \mathcal{G} (independent of points in Y due to uniformity) and \delta_X = \delta_Y using the uniformity of \mathcal{F} to ensure d_Y(f(x), f(y)) < \delta_Y implies the composed differences remain below \epsilon.^[7] An equicontinuous family \mathcal{F} \subset C(X, Y) maps compact subsets of X to totally bounded subsets of Y. For a compact K \subset X, cover K with finitely many \delta-balls where \delta is from equicontinuity for \epsilon = 1/n, and the images under each f \in \mathcal{F} form a finite \epsilon-net in f(K), yielding a countable \epsilon-net for \bigcup_{f \in \mathcal{F}} f(K).^[7]

Examples and Applications

Illustrative Examples

A fundamental example of an equicontinuous family is the set of all constant functions on a metric space (X, d). For any constant function f(x) = c where c \in \mathbb{R}, the difference satisfies |f(x) - f(y)| = 0 < \epsilon for every \epsilon > 0 and any x, y \in X, regardless of the choice of \delta > 0. Thus, the family is equicontinuous at every point, with no restriction on the size of the family. This trivial case illustrates how equicontinuity holds when functions exhibit no variation, which is relevant in studying limits of constant sequences in functional analysis. A more general class consists of families of Lipschitz continuous functions with a uniform Lipschitz constant. Consider the family \mathcal{F} = \{ f: X \to \mathbb{R} \mid |f(x) - f(y)| \leq K d(x, y) \ \forall x, y \in X \} for a fixed K > 0. For any \epsilon > 0, choosing \delta = \epsilon / K ensures |f(x) - f(y)| \leq K d(x, y) < K \delta = \epsilon for all f \in \mathcal{F}, uniformly across the family. Such families are equicontinuous on any domain and play a key role in proving compactness in spaces of continuous functions, such as via the Arzelà–Ascoli theorem.^[8] On the real line \mathbb{R}, the family of translations \{ f_a(x) = x + a \mid a \in A \} where A \subset \mathbb{R} is bounded provides another illustration. Here, |f_a(x) - f_a(y)| = |x - y| for all a \in A, so the difference is independent of the parameter a. Given \epsilon > 0, set \delta = \epsilon; then |f_a(x) - f_a(y)| < \epsilon whenever |x - y| < \delta, uniformly for all a. Boundedness of A ensures the family remains controlled, though equicontinuity holds even without it due to the uniform bound on differences; this example demonstrates preservation under parameter shifts in dynamical systems and approximation theory. Polynomials of bounded degree on a compact interval also form equicontinuous families when uniformly bounded. Let \mathcal{P}_n(I) = \{ p: I \to \mathbb{R} \mid \deg(p) \leq n, \|p\|_\infty \leq M \} where I \subset \mathbb{R} is compact and M > 0 is fixed. Each such polynomial is continuously differentiable with derivative of degree at most n-1, and the uniform bound implies the derivatives are uniformly bounded by Markov's inequality or direct estimation, yielding a uniform Lipschitz constant. Thus, for \epsilon > 0, \delta = \epsilon / L with L depending on n, M, |I| works for all p \in \mathcal{P}_n(I). This equicontinuity is crucial for Weierstrass approximation, enabling uniform limits of polynomials to continuous functions on compacts.^[9] Finally, consider the family \{ \sin(\omega x) \mid \omega \in [0, M] \} on \mathbb{R} for fixed M > 0. By the mean value theorem, |\sin(\omega x) - \sin(\omega y)| = |\cos(\xi)| \cdot |\omega| \cdot |x - y| \leq M |x - y| for some \xi between \omega x and \omega y. Choosing \delta = \epsilon / M ensures the difference is less than \epsilon uniformly over the family. The compact frequency range prevents oscillations from becoming arbitrarily rapid, making this family equicontinuous and useful in Fourier analysis for bounded-spectrum signals.

Counterexamples

A classic counterexample to equicontinuity is the family of functions \{f_n(x) = n x \mid n \in \mathbb{N}\} defined on the compact interval [0,1]. Each f_n is continuous on [0,1], as it is a linear function. However, the family fails to be equicontinuous at x = [0](/page/0). To see this, fix \epsilon = [1](/page/1). For any \delta > 0, choose n > 1/\delta; then |f_n(\delta) - f_n(0)| = n \delta > 1 = \epsilon. Thus, no single \delta > 0 works uniformly for all n, and the required \delta_n = \epsilon / n \to 0 as n \to \infty. This illustrates the necessity of bounded variation across the family, as the slopes grow unboundedly. Another family that fails equicontinuity everywhere on [0,1] is \{f_n(x) = n^2 x (1 - x)^n \mid n \in \mathbb{N}\}. Each f_n is a polynomial, hence continuous on [0,1]. Near x = 0, f_n(x) \approx n^2 x, which exhibits rapid growth similar to the previous example, preventing a uniform modulus of continuity at 0. Moreover, the maximum value of f_n occurs near x = 1/n and is approximately n / e, confirming unbounded slopes locally across the interval. Consequently, for any point x_0 \in [0,1] and \epsilon > 0, no \delta > 0 bounds |f_n(x) - f_n(x_0)| for all n when |x - x_0| < \delta, due to the oscillatory amplification near boundaries. This example underscores the role of compactness in controlling such rapid local changes. Consider the parametrized family \{f_a(x) = x / a \mid a > 0\} on the non-compact interval (0,1]. Each f_a is continuous on (0,1], being linear with positive slope $1/a. Yet, the family is not equicontinuous at any point x_0 \in (0,1]. For fixed \epsilon > 0 and x_0, |f_a(x) - f_a(x_0)| = |x - x_0| / a. To ensure this is less than \epsilon for all a > 0 whenever |x - x_0| < \delta, one would need \delta / a < \epsilon for all a > 0, which forces \delta = 0 as a \to 0^+. The modulus of continuity deteriorates uniformly with decreasing a, highlighting the need for bounded parameters in the family to achieve equicontinuity. On the non-compact domain \mathbb{R}, the family of translations \{f_a(x) = x + a \mid a \in \mathbb{R}\}, with a unbounded, illustrates the interplay between equicontinuity and uniform boundedness. Each f_a is continuous on \mathbb{R}, and the family is equicontinuous: |f_a(y) - f_a(x_0)| = |y - x_0| for all a, so \delta = \epsilon works uniformly independent of a and points. However, the family lacks uniform boundedness, as \sup_{x \in \mathbb{R}} |f_a(x)| = \infty for each a, showing why boundedness is essential alongside equicontinuity for relative compactness.

Equicontinuous Linear Operators

Characterizations

In the context of linear operators between normed spaces, a family of continuous linear operators \{T_i : X \to Y\} is equicontinuous if and only if it is uniformly bounded, meaning there exists a constant M > 0 such that \sup_i \|T_i x\| \leq M \|x\| for all x \in X, or equivalently, \sup_{\|x\| \leq 1} \|T_i x\| < \infty uniformly over i. This equivalence stems from the homogeneity and additivity of linear operators, which allow the modulus of continuity to be controlled globally by its behavior on the unit ball. The Banach–Steinhaus theorem provides a fundamental characterization relating pointwise boundedness to equicontinuity for families of continuous linear operators from a Banach space X to a normed space Y: if the family is pointwise bounded, meaning \sup_i \|T_i x\| < \infty for each fixed x \in X, then it is equicontinuous (and hence uniformly bounded). This result, also known as the uniform boundedness principle, relies on the completeness of X and highlights how pointwise control implies uniform operator norm control in Banach space settings.^[10] For families of linear functionals, where Y = \mathbb{R} or \mathbb{C}, equicontinuity is equivalent to uniform boundedness in the dual norm: there exists M > 0 such that |\phi(x)| \leq M \|x\| for all \phi in the family and all x \in X. This special case follows directly from the general linear operator equivalence, as functionals are linear maps to the scalar field. Unlike the general case for arbitrary functions, where equicontinuity must be verified at every point, linearity ensures that equicontinuity at the origin $0 \in X implies global equicontinuity, due to the scaling property T(\lambda x) = \lambda T(x).

Key Properties

Equicontinuous families of linear operators between normed linear spaces are closed under addition and scalar multiplication. Specifically, if \{T_\alpha : X \to Y\} is an equicontinuous family of bounded linear operators and \lambda \in \mathbb{K} (where \mathbb{K} is the scalar field), then the family \{\sum c_\beta T_\beta : X \to Y\}, consisting of finite linear combinations, remains equicontinuous. This follows from the fact that equicontinuity is equivalent to uniform boundedness for such families, and the set of uniformly bounded operators forms a vector space.^[11] A key characterization links pointwise boundedness to equicontinuity: for a family F of continuous linear functionals on a Banach space X, if \sup_{f \in F} |f(x)| < \infty for every x \in X, then F is equicontinuous. This is a direct consequence of the uniform boundedness principle (Banach-Steinhaus theorem), which ensures that pointwise boundedness on the Banach space implies uniform boundedness in operator norm, hence equicontinuity at the origin and thus everywhere.^[12] Adjoint operators preserve equicontinuity. If \{T_\alpha : X \to Y\} is an equicontinuous family of bounded linear operators between Banach spaces X and Y, then the adjoint family \{T_\alpha^* : Y^* \to X^*\} is also equicontinuous. This holds because the operator norm satisfies \|T_\alpha^*\| = \|T_\alpha\| for each \alpha, so uniform boundedness transfers to the adjoints.^[11]^[12] For families of functionals, equicontinuity implies relative compactness in the weak* topology when X is a Banach space. An equicontinuous subset F \subset X^* is relatively compact in the weak* topology \sigma(X^*, X), as it is uniformly bounded and thus contained in a scalar multiple of the closed unit ball, which is weak* compact by the Banach-Alaoglu theorem.^[12]^[11] Equicontinuous linear operators preserve completeness under surjectivity. If T : X \to Y is a bounded linear operator (hence equicontinuous as a singleton family) between normed spaces, with X complete and T surjective, then Y is complete. This results from the open mapping theorem, which guarantees that surjective bounded operators between Banach spaces are open mappings, transferring completeness from the domain to the codomain.^[12]

Connections to Convergence

Uniform Convergence

A fundamental connection between equicontinuity and convergence arises in the context of sequences or families of functions on compact metric spaces. If a sequence of continuous functions \{f_n\} on a compact set K is equicontinuous and converges pointwise to a function f, then the convergence is uniform on K, and f is continuous.^[7] This result highlights how equicontinuity strengthens pointwise convergence to ensure uniformity, preventing pathological behaviors where the limit might fail to preserve desirable properties like continuity. Equicontinuous and pointwise bounded families on compact sets also admit uniformly convergent subsequences, as guaranteed by compactness criteria in the space of continuous functions equipped with the uniform norm.^[7] For instance, on a compact interval, an equicontinuous family that is bounded pointwise possesses a subsequence converging uniformly to a continuous limit function. This interplay underscores equicontinuity's role in facilitating controlled convergence, extending beyond monotonic cases like Dini's theorem to general settings. In contrast, sequences lacking equicontinuity may converge pointwise without achieving uniform convergence. A classic counterexample is the sequence f_n(x) = x^n on the compact interval [0,1], which converges pointwise to the discontinuous function f(x) = 0 for

x \in [0,1)&#36; and

f(1) = 1, but the supremum norm |f_n - f|_\infty = 1does not tend to zero, confirming non-uniform convergence; moreover,{f_n}is not equicontinuous nearx=1$.^[13] Uniform convergence preserves continuity: if each f_n is continuous on a set and \{f_n\} converges uniformly to f, then f is continuous. For equicontinuous sequences, the uniform limit not only inherits continuity but also ensures that adjoining the limit to the original family maintains equicontinuity, as the uniform Cauchy property aligns with the shared modulus of continuity across the sequence.^[7] Equicontinuity can be viewed through the lens of a uniform modulus of continuity across the family. Specifically, for every \varepsilon > 0, there exists \delta > 0 (independent of the functions in the family) such that |x - y| < \delta implies |f(x) - f(y)| < \varepsilon for all f in the family; this boundedness of \delta(\varepsilon, f) over f distinguishes equicontinuous families and underpins their convergence properties.^[7]

Arzelà–Ascoli Theorem

The Arzelà–Ascoli theorem provides a characterization of relatively compact subsets in the space of continuous functions C(K, \mathbb{R}), where K is a compact metric space, equipped with the topology of uniform convergence. A subset F \subseteq C(K, \mathbb{R}) is relatively compact if and only if F is pointwise bounded and equicontinuous.^[14] Pointwise boundedness means that for every x \in K, the set \{f(x) : f \in F\} is bounded in \mathbb{R}, while equicontinuity requires that for every \varepsilon > 0, there exists \delta > 0 such that |f(x) - f(y)| < \varepsilon for all f \in F whenever d_K(x, y) < \delta.^[14] This criterion ensures the existence of uniformly convergent subsequences from sequences in F, making it a cornerstone for compactness in function spaces.^[1] The theorem was originally established by Giulio Ascoli in 1883–1884 for real-valued functions on compact intervals, focusing on equicontinuous and bounded families, and generalized by Cesare Arzelà in 1889 to the broader setting of continuous functions.^[15] A proof sketch proceeds in two directions. For the necessity, relative compactness in the uniform topology implies uniform boundedness (hence pointwise boundedness) and equicontinuity, as limits of uniformly convergent sequences preserve these properties on compact domains.^[1] For sufficiency, equicontinuity implies that F is totally bounded in the uniform metric: cover K with finitely many balls of radius \delta corresponding to \varepsilon/3, and use pointwise boundedness to select a finite \varepsilon/3-net in \mathbb{R} for each center, yielding a finite uniform net for F.^[14] Combined with completeness of the closure (via uniform limits), this shows relative compactness. Alternatively, boundedness and equicontinuity enable the Arzelà diagonal argument: extract a pointwise convergent subsequence on a countable dense subset of K, then use equicontinuity to extend uniform convergence to all of K.^[1] Extensions of the theorem accommodate broader settings. For codomains Y that are Banach spaces, the result holds if F is uniformly equicontinuous and uniformly bounded, ensuring the diagonal subsequence converges uniformly.^[16] Infinite-dimensional versions apply directly to C(K, B) where B is a Banach space, with relative compactness equivalent to uniform boundedness and equicontinuity, leveraging the theorem's metric foundation via the norm topology on B.^[16] A key application arises in the existence theory for ordinary differential equations (ODEs). For the initial value problem y' = f(t, y) with continuous f on a compact rectangle, the associated integral operator maps bounded sets into equicontinuous and pointwise bounded families in C([a,b], \mathbb{R}^n); Arzelà–Ascoli then guarantees a fixed point via compactness, proving Peano's existence theorem without uniqueness.^[17] This compact embedding approach extends to nonlinear problems, establishing solution existence through sequential compactness.^[17]

Advanced Generalizations

Topological Spaces

In topological spaces, the notion of equicontinuity generalizes the metric case by replacing ε-balls with neighborhoods. Consider a family \mathcal{F} of functions from a topological space X to a topological space Y. The family \mathcal{F} is said to be equicontinuous at a point x \in X if for every neighborhood V of f(x) in Y (for any f \in \mathcal{F}), there exists a neighborhood U of x in X such that f(U) \subset V for all f \in \mathcal{F}. The family \mathcal{F} is equicontinuous if it is equicontinuous at every point of X.^[18] This definition aligns with the metric version when Y is metrizable, as neighborhoods can be taken as metric balls. A key property is that equicontinuity implies individual continuity: for each f \in \mathcal{F}, the condition reduces to the standard definition of continuity at x, since the neighborhood U works uniformly across the family. In the more structured setting of uniform spaces, where Y is equipped with a uniformity generating its topology, equicontinuity can be reformulated using entourages. Specifically, for an entourage V in Y, there exists a neighborhood U of x such that (f(x), f(y)) \in V for all y \in U and all f \in \mathcal{F}; the sets V[f(x)] = \{ z \in Y \mid (f(x), z) \in V \} then serve as neighborhoods of f(x). This entourage-based view is equivalent to the neighborhood definition when the uniformity is compatible with the topology, and it extends naturally to uniform equicontinuity across the entire space X, where a single entourage U in X works for every V in Y. In uniform spaces, an equicontinuous family on a compact domain is uniformly equicontinuous.^[18] A significant application arises in compactness theorems for function spaces. In the setting of uniform spaces, a family \mathcal{F} \subset C(X, Y) of continuous functions, where X is a locally compact topological space and Y is a Hausdorff uniform space, is relatively compact in the topology of uniform convergence if and only if it is equicontinuous and pointwise relatively compact (meaning the image \mathcal{F}(x) is relatively compact in Y for each x \in X), provided \mathcal{F} satisfies an additional extension property related to the uniformity on Y. This generalizes the classical Arzelà–Ascoli theorem beyond metric spaces.^[18] Equicontinuity plays a crucial role in non-metrizable spaces, such as the space of continuous functions C(X, Y) equipped with the compact-open topology, where X is compact but Y may not be metrizable. In this topology, subbasic open sets are of the form \{ f \in C(X, Y) \mid f(K) \subset V \} for compact K \subset X and open V \subset Y. Equicontinuous families ensure that pointwise limits remain continuous and preserve relative compactness, facilitating the study of infinite-dimensional topological vector spaces or spaces of maps between non-metrizable manifolds. For instance, in the compact-open topology on C(\mathbb{R}, \mathbb{R}), equicontinuous subsets correspond to families that are "uniformly controlled" near compact sets, highlighting behaviors not captured in metrizable settings.

Stochastic Equicontinuity

Stochastic equicontinuity adapts the concept of equicontinuity to families of random functions or stochastic processes, incorporating probabilistic measures to handle almost-sure or in-probability uniformity. A family of random functions \{f_\omega : \omega \in \Omega\} defined on a metric space is almost surely equicontinuous if, for almost every \omega, the realized path f_\omega is equicontinuous in the deterministic sense, meaning that for every \epsilon > 0, there exists \delta > 0 such that |f_\omega(x) - f_\omega(y)| < \epsilon whenever |x - y| < \delta. An alternative formulation emphasizes convergence in probability: for every \epsilon > 0,

\lim_{\delta \to 0} \sup_{\omega} P\left( \sup_{|x-y| < \delta} |f(\omega, x) - f(\omega, y)| > \epsilon \right) = 0,

where the supremum over \omega may be restricted to a suitable index set for uniformity across the family. This probabilistic strengthening ensures control over path regularity in random settings, distinguishing it from purely topological equicontinuity by accounting for measure-theoretic properties.^[19]^[20] In applications to empirical processes, stochastic equicontinuity plays a central role in establishing uniform convergence theorems. Glivenko–Cantelli classes of functions—those for which the supremum norm of the difference between the empirical measure and the true probability measure converges to zero almost surely—are characterized by stochastic equicontinuity of the associated centered empirical process. Specifically, for a class \mathcal{F} of measurable functions, stochastic equicontinuity implies that the process \sqrt{n} (P_n - P) remains tight and converges uniformly, enabling the Glivenko–Cantelli theorem to hold and facilitating consistency of empirical risk minimizers in statistical learning. This connection underscores how stochastic equicontinuity bridges pointwise probabilistic limits to uniform behavior over function classes.^[21] The Kolmogorov–Chentsov theorem provides a foundational result for stochastic equicontinuity in Gaussian processes, linking moment conditions on increments to sample path regularity. For a stochastic process \{X_t : t \in [0,1]\} satisfying E[|X_t - X_s|^\alpha] \leq C |t - s|^{1 + \beta} for constants C > 0, \alpha > 0, and \beta > 0, the theorem guarantees the existence of a continuous modification whose paths are almost surely Hölder continuous with exponent \gamma for any \gamma < \beta / \alpha. This Hölder continuity implies stochastic equicontinuity on compact intervals, as the uniform modulus of continuity bounds oscillations almost surely. For Gaussian processes, such as those with stationary increments, these conditions ensure that sample paths are almost surely equicontinuous, providing a quantitative basis for pathwise analysis in stochastic modeling.^[22] Within Vapnik–Chervonenkis (VC) theory, entropy bounds on function classes directly imply stochastic equicontinuity, supporting uniform convergence in high-dimensional statistics. A class \mathcal{F} with finite VC dimension d has entropy \log N(\epsilon, \mathcal{F}, L_2(P)) \lesssim d \log(1/\epsilon), where N denotes the covering number; this polynomial growth ensures that the empirical process indexed by \mathcal{F} is stochastically equicontinuous, as chaining arguments bound the expected supremum via entropy integrals. Such classes are thus Glivenko–Cantelli, with applications to bounding generalization error in machine learning. Seminal work by Vapnik and Chervonenkis established these entropy conditions as sufficient for stochastic equicontinuity, influencing modern statistical theory.^[21]^[23] A representative example is the standard Brownian motion \{W_t : t \in [0,1]\}, whose sample paths are almost surely equicontinuous via the Kolmogorov–Chentsov theorem. The increments satisfy E[|W_t - W_s|^4] = |t - s|^2, fitting the theorem with \alpha = 4 and \beta = 1, yielding paths that are almost surely Hölder continuous for exponents less than $1/4. This modulus of continuity, \sup_{|t-s| < \delta} |W_t - W_s| = O(\delta^{1/2} (\log(1/\delta))^{1/2}) almost surely, confirms equicontinuity on [0,1], illustrating how Gaussian properties yield pathwise uniformity essential for stochastic calculus.^[22]

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Counterexamples in analysis / Bernard R. Gelbaum, John M.H. Olmsted. p. cm. . an unabridged, slightly corrected republication of the 1965 second.
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Théorie des opérations linéaires - EuDML
How to cite ... Stefan Banach. Théorie des opérations linéaires. 1932. <http://eudml.org/doc/268537>. @book{StefanBanach1932, abstract = {PRÉFACE..Access Full Book · Abstract · How to cite
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Sur le principe de la condensation de singularités - EuDML
Sur le principe de la condensation de singularités. Stefan Banach; Hugo Steinhaus · Fundamenta Mathematicae (1927). Volume: 9, Issue: 1, page 50-61 ...
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[PDF] FUNCTIONAL ANALYSIS - ETH Zürich
Jun 8, 2017 · These are notes for the lecture course “Functional Analysis I” held by the second author at ETH Zürich in the fall semester 2015.
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[PDF] Functional Analysis, Sobolev Spaces and Partial Differential Equations
The first part deals with abstract results in FA and operator theory. The second part concerns the study of spaces of functions (of one or more real variables) ...
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Math 521 Uniform Convergence
For example, the sequence fn(x)=xn from the previous example converges pointwise on the interval [0,1], but it does not converge uniformly on this interval.
[20]
[PDF] The Arzel`a-Ascoli Theorem
Feb 20, 2017 · Theorem 1. Let (T,d) be a compact metric space. Then F⊂C(T,R) is relatively compact if and only if it is uniformly equicontinuous and for ...
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An improvement of the Arzela–Ascoli theorem - ResearchGate
Aug 6, 2025 · ... In 1884, Ascoli proved the prototype of Ascoli's theorem in [3], while Arzelà proved it independently in [2], acknowledging Ascoli's ...
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[PDF] ASCOLI-ARZEL`A THEOREM Theorem. If K is a compact metric ...
The definition of equicontinuity of a subset of C(K) is 'uniform (in f ∈ F) uniform (in the point in K) continuity'. That is, given > 0 there must exist δ ...Missing: mathematics | Show results with:mathematics
[23]
[PDF] Notes about C(K), Arzel`a-Ascoli and applications to differential ...
Now we show the equicontinuity of the family F. By a Lemma above each fi is uniformly continuous. Thus for each i there exists a δi > 0 such that. |fi(x) ...
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[1602.05693] Arzelà-Ascoli theorem in uniform spaces - arXiv
Feb 18, 2016 · The main part of the paper introduces the notion of extension property which, similarly as equicontinuity, equates different topologies on C(X,Y) ...Missing: original | Show results with:original
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[PDF] Uniform Convergence in Probability and Stochastic Equicontinuity
It is well known that pointwise convergence and equicontinuity characterize uniform convergence to a continuous function on a compact set; e.g. see Rudin (1976, ...
[26]
[PDF] arXiv:1206.2385v2 [math.ST] 11 Jun 2013
Jun 11, 2013 · Stochastic equicontinuity typically captures the key difficulty in weak convergence proofs of estimators with non-differentiable objective ...
[27]
[PDF] A Gentle Introduction to Empirical Process Theory and Applications
3 Glivenko-Cantelli (GC) classes of functions. Suppose that X1,...,Xn are independent random variables defined on the space X with probability measure P. Let ...
[28]
[PDF] gaussian processes; kolmogorov-chentsov theorem
2.1. Continuity of Sample Paths. The Kolmogorov-Chentsov theorem provides a useful criterion for establishing the existence of versions of stochastic processes ...Missing: equicontinuity | Show results with:equicontinuity
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[PDF] Spring 2018 Statistics 210b (Theoretical Statistics) - All Lectures
Jan 16, 2018 · is via the Vapnik Chervonenkis dimension (or simply the VC ... K, we need to prove stochastic equicontinuity for which we need to bound.