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Uniform norm

The uniform norm, also known as the supremum norm or sup norm, is a fundamental concept in functional analysis and real analysis that measures the maximum deviation of a function from zero over its domain.^[1] For a bounded function f: X \to \mathbb{R} (or \mathbb{C}) defined on a set X, it is defined as \|f\|_\infty = \sup_{x \in X} |f(x)|, where the supremum is the least upper bound of the absolute values of f.^[2] This norm quantifies the "size" of f in a uniform sense across the entire domain, distinguishing it from integral-based norms like the L^p norms.^[3] In the context of continuous functions on a compact metric space X, the uniform norm equips the space C(X) of all continuous real- or complex-valued functions on X with a complete norm, making C(X) a Banach space.^[4] This structure is essential for studying uniform convergence of sequences of functions, where a sequence \{f_n\} converges uniformly to f if \|f_n - f\|_\infty \to 0 as n \to \infty, preserving properties like continuity and integrability under limits.^[1] The uniform norm also plays a key role in approximation theory, such as the Stone-Weierstrass theorem, which guarantees dense polynomial approximations in C(X) under this norm.^[3] Beyond function spaces, the uniform norm extends to vector-valued functions and finite-dimensional spaces, where for a vector \mathbf{x} = (x_1, \dots, x_n) \in \mathbb{R}^n, it is \|\mathbf{x}\|_\infty = \max_{1 \leq i \leq n} |x_i|, inducing a metric for convergence and continuity analysis.^[5] Its properties include submultiplicativity in certain algebras and equivalence to other norms up to constants in finite dimensions, ensuring consistent topological behavior.^[6] Applications span numerical analysis, where it bounds errors in approximations, and operator theory, measuring the norm of bounded linear operators on normed spaces.^[4]

Definition and Basic Concepts

Formal Definition

The uniform norm, also known as the supremum norm or sup norm, is defined for a scalar-valued function f: X \to \mathbb{R} (or \mathbb{C}) on a set X by

\|f\|_\infty = \sup \{ |f(x)| : x \in X \},

where the supremum is taken in the extended real numbers [0, \infty], and the norm is finite if and only if f is bounded on X.^[7] This definition equips the space of all bounded functions from X to \mathbb{R} (or \mathbb{C}) with a norm that measures the maximum deviation of |f(x)| over the domain.^[4] For vector-valued functions f: X \to Y, where Y is a normed vector space with norm \|\cdot\|_Y, the uniform norm extends naturally as

\|f\|_\infty = \sup \{ \|f(x)\|_Y : x \in X \},

again allowing values in [0, \infty] and requiring boundedness of f for finiteness.^[8] In this setting, the uniform norm operates on the space Y^X of all functions from X to Y, serving as an extended norm that assigns \infty to unbounded functions, thereby distinguishing bounded mappings as those with finite norm.^[7] In common applications, such as the space of continuous functions C(K) on a compact set K, the uniform norm is always finite because continuous functions on compact sets are bounded, ensuring the supremum is attained as a maximum.^[4] Similarly, for bounded continuous functions on non-compact domains, like C_b(\mathbb{R}), the norm remains finite by the boundedness assumption.^[7]

Examples in Vector Spaces and Function Spaces

In finite-dimensional vector spaces, the uniform norm, also known as the max norm or Chebyshev norm, is defined for a vector \mathbf{x} = (x_1, \dots, x_n) \in \mathbb{R}^n as \|\mathbf{x}\|_\infty = \max_{1 \leq i \leq n} |x_i|.^[9] This norm captures the largest absolute component of the vector, providing a measure of its "size" based on the extremal value rather than an average. For instance, in \mathbb{R}^3, the vector (1, -2, 3) has \|(1, -2, 3)\|_\infty = 3.^[9] In the space of continuous functions C[a, b] on a compact interval [a, b], the uniform norm is given by \|f\|_\infty = \sup_{x \in [a, b]} |f(x)|.^[3] Since f is continuous on the compact set [a, b], it attains its maximum value, so the supremum equals the maximum: \|f\|_\infty = \max_{x \in [a, b]} |f(x)|. This norm quantifies the maximum deviation of f from zero over the interval, emphasizing uniform boundedness across the domain. The space \ell^\infty of bounded real sequences (a_n)_{n \in \mathbb{N}} is equipped with the uniform norm \|(a_n)\|_\infty = \sup_{n \in \mathbb{N}} |a_n|.^[10] This extends the finite-dimensional concept to infinite sequences, where the norm is the least upper bound of the absolute values, ensuring the sequence remains bounded. A concrete computation in C[0, 1] is the function f(x) = x^2, for which \|f\|_\infty = \sup_{x \in [0, 1]} |x^2| = 1, attained at x = 1.

Topology Induced by the Uniform Norm

Uniform Metric

The uniform metric, also known as the supremum metric or \ell^\infty metric, is induced by the uniform norm on spaces of functions or sequences. For functions f, g: X \to \mathbb{R}, where X is a set and the supremum exists, it is defined as

d_\infty(f, g) = \|f - g\|_\infty = \sup_{x \in X} |f(x) - g(x)|.

^[11] This metric extends naturally to functions taking values in a normed space Y, replacing the absolute value with the norm in Y.^[11] As a metric, d_\infty satisfies the standard axioms: non-negativity (d_\infty(f, g) \geq 0), symmetry (d_\infty(f, g) = d_\infty(g, f)), the triangle inequality (d_\infty(f, h) \leq d_\infty(f, g) + d_\infty(g, h)), and d_\infty(f, g) = 0 if and only if f = g pointwise.^[11] These properties follow from the corresponding properties of the absolute value and the supremum operation, ensuring it measures the maximum deviation between functions across the domain. In finite-dimensional spaces, the uniform metric coincides with the Chebyshev distance. For vectors x, y \in \mathbb{R}^n, d_\infty(x, y) = \max_{1 \leq i \leq n} |x_i - y_i|, which quantifies the largest coordinate-wise difference.^[11] The uniform metric plays a key role in establishing completeness for certain function spaces. Notably, the space C[a, b] of continuous real-valued functions on the compact interval [a, b], equipped with d_\infty, forms a complete metric space, meaning every Cauchy sequence converges to a continuous function in the space.^[11] In the context of the uniform norm, also known as the supremum norm \| \cdot \|_\infty, uniform convergence of a sequence of functions \{f_n\} to a function f on a domain X is defined such that \|f_n - f\|_\infty \to 0 as n \to \infty. This condition is equivalent to \sup_{x \in X} |f_n(x) - f(x)| \to 0, meaning the maximum deviation between f_n and f uniformly diminishes across the entire domain, independent of the specific point x.^[12] This mode of convergence is stricter than pointwise convergence, where the limit is approached at each point individually, potentially at varying rates; uniform convergence ensures global control via the sup-norm, preserving properties like continuity of the limit function under suitable conditions.^[12] The uniform norm induces a topology on the space of bounded functions from X to \mathbb{R} (or \mathbb{C}), often denoted B(X), where the open sets are arbitrary unions of open balls B(f, \varepsilon) = \{g \in B(X) : \|g - f\|_\infty < \varepsilon\} for f \in B(X) and \varepsilon > 0. This norm topology, equivalent to the topology of uniform convergence, equips B(X) with a metric structure that captures uniform closeness between functions.^[13] Sequences converge in this topology precisely when they converge uniformly, making it a natural framework for studying limits in function spaces.^[13] A weaker but equivalent uniform structure underlies this topology, generated by the base of entourages U_\varepsilon = \{(f, g) \in B(X) \times B(X) : \|f - g\|_\infty < \varepsilon\} for all \varepsilon > 0. These entourages define a uniformity on B(X) that induces the same topology as the norm metric, providing an entourage-based perspective on uniform convergence without relying explicitly on distances.^[14] This uniform structure arises in the broader theory of uniform spaces, where it corresponds to the uniformity of uniform convergence on the full domain X, distinguishing it from pointwise or compact-open uniformities on related function spaces.^[14]

Properties of the Uniform Norm

Algebraic and Analytic Properties

The uniform norm, defined as \|f\|_\infty = \sup_{x \in D} |f(x)| for a function f on a domain D, satisfies the axioms of a norm on appropriate vector spaces of functions. Specifically, positivity holds: \|f\|_\infty \geq 0 for all f, with equality if and only if f = 0 everywhere on D, since the supremum of non-negative values is zero only when |f(x)| = 0 for all x.^[15] Homogeneity is verified by \|\alpha f\|_\infty = |\alpha| \|f\|_\infty for scalar \alpha, as \sup |\alpha f(x)| = |\alpha| \sup |f(x)|.^[15] The triangle inequality \|f + g\|_\infty \leq \|f\|_\infty + \|g\|_\infty follows from |f(x) + g(x)| \leq |f(x)| + |g(x)| \leq \|f\|_\infty + \|g\|_\infty for all x, so the supremum respects this bound.^[15] In spaces of bounded functions equipped with pointwise multiplication, such as the continuous functions C[a,b] on a closed interval, the uniform norm is submultiplicative: \|fg\|_\infty \leq \|f\|_\infty \|g\|_\infty. This arises because |f(x)g(x)| \leq |f(x)| |g(x)| \leq \|f\|_\infty \|g\|_\infty for all x \in [a,b], ensuring the supremum of the product is controlled by the product of suprema.^[16] The space C(K) of continuous real- or complex-valued functions on a compact set K, normed by \|\cdot\|_\infty, is complete, making it a Banach space. Every Cauchy sequence in C(K) converges uniformly to a continuous function on K, preserving the norm's completeness.^[17] By definition, \|f\|_\infty < \infty if and only if f is bounded on its domain, as the supremum exists and is finite precisely when |f(x)| remains below some bound for all x.^[3]

Geometric Interpretations

In finite-dimensional Euclidean space \mathbb{R}^n, the unit ball under the uniform norm, defined as \{x = (x_1, \dots, x_n) \in \mathbb{R}^n : \|x\|_\infty \leq 1\} where \|x\|_\infty = \max_{1 \leq i \leq n} |x_i|, forms a hypercube with side length 2 centered at the origin, specifically the set [-1, 1]^n. This geometric shape arises because the norm constrains each coordinate independently to lie within [-1, 1], resulting in flat faces parallel to the coordinate hyperplanes and vertices at all points of the form (\pm 1, \dots, \pm 1).^[18] The volume of this hypercube is $2^n, which increases exponentially with the dimension n, standing in sharp contrast to the unit ball under the Euclidean norm, whose volume \pi^{n/2} / \Gamma(n/2 + 1) approaches zero as n \to \infty. This difference highlights how the uniform norm emphasizes the maximum deviation in any direction, leading to a "boxy" geometry that fills more of the space compared to the increasingly "spiky" Euclidean ball in high dimensions.^[19] In infinite-dimensional function spaces, such as the space C[a, b] of continuous functions on a compact interval [a, b] equipped with the uniform norm \|f\|_\infty = \sup_{x \in [a, b]} |f(x)|, the unit ball consists of all functions f satisfying |f(x)| \leq 1 for every x \in [a, b]. Geometrically, this can be visualized as an infinite-dimensional "tube" of radius 1 surrounding the x-axis in the graph of functions, where the boundary comprises functions that touch \pm 1 at some points while remaining within the bounds elsewhere, reflecting the norm's focus on pointwise supremum control.^[3] The dual norm of the uniform norm \|\cdot\|_\infty on \mathbb{R}^n is the \ell^1 norm \|\cdot\|_1, defined by \|y\|_1 = \sum_{i=1}^n |y_i|. Consequently, the unit ball of the dual norm is the cross-polytope (or \ell^1 ball), a polytope with $2^n facets and vertices at the standard basis vectors scaled by \pm 1, such as (\pm 1, 0, \dots, 0) and permutations. This duality underscores the geometric complementarity: the hypercube's faces correspond to the cross-polytope's vertices, illustrating how the uniform norm's geometry pairs with the summation-based structure of its dual.^[20]

Relations to Other Norms and Structures

Comparisons with p-Norms

The uniform norm, also known as the L^\infty norm, exhibits a close relationship with L^p norms through limiting behavior as p \to \infty. For a measurable function f \in L^r(X, \mu) where r < \infty and \mu(X) < \infty, it holds that

\lim_{p \to \infty} \|f\|_p = \|f\|_\infty,

where \|f\|_p = \left( \int_X |f|^p \, d\mu \right)^{1/p} and \|f\|_\infty = \esssup_{x \in X} |f(x)|. This convergence arises from applying Chebyshev's inequality to bound the measure of sets where |f| exceeds certain thresholds, establishing both \liminf_{p \to \infty} \|f\|_p \geq \|f\|_\infty and \limsup_{p \to \infty} \|f\|_p \leq \|f\|_\infty.^[21] On finite measure spaces, the uniform norm relates to L^p norms via inclusion and boundedness properties. Specifically, for $1 \leq p < \infty, the space L^\infty(X, \mu) embeds continuously into L^p(X, \mu), meaning L^\infty \subset L^p with

\|f\|_p \leq \mu(X)^{1/p} \|f\|_\infty

for all f \in L^\infty(X, \mu). This follows from Hölder's inequality applied to the constant function 1, yielding \int_X |f|^p \, d\mu \leq \|f\|_\infty^p \mu(X). Moreover, on probability spaces where \mu(X) = 1, the L^p norms are monotone in p, satisfying \|f\|_p \leq \|f\|_q for $1 \leq p \leq q < \infty, which extends to \|f\|_p \leq \|f\|_\infty. These relations highlight how the uniform norm captures the "worst-case" supremum behavior, contrasting with the integral averaging of L^p norms.^[21]^[22] In finite-dimensional settings, such as vectors in \mathbb{R}^n or \mathbb{C}^n equipped with the counting measure (where \mu(X) = n), the inequalities take a precise form analogous to the functional case. For $1 \leq p < \infty,

\|x\|_\infty \leq \|x\|_p \leq n^{1/p} \|x\|_\infty.

The left inequality holds because the p-norm exceeds the maximum component: \sum_{i=1}^n |x_i|^p \geq \max_i |x_i|^p, so \|x\|_p \geq \|x\|_\infty. The right inequality follows from bounding each |x_i| \leq \|x\|_\infty, yielding \sum_{i=1}^n |x_i|^p \leq n \|x\|_\infty^p. Equality in the left holds for vectors with a single nonzero entry (standard basis vectors), while equality in the right occurs for vectors where all components equal the maximum in absolute value (e.g., the all-ones vector). These bounds demonstrate the equivalence of norms in finite dimensions, with the factor n^{1/p} vanishing as p \to \infty.^[23] The embedding L^\infty \subset L^p for $1 \leq p < \infty on probability spaces underscores the dominance of the uniform norm: functions bounded almost everywhere are p-integrable, but the converse fails, as L^p contains unbounded functions. This inclusion, combined with the norm bounds, implies that the uniform norm provides a stricter control, essential for uniform convergence and supremum estimates in analysis.^[22]

Embeddings and Approximation Theorems

The space of continuous real-valued functions C(K) on a compact Hausdorff space K, equipped with the uniform norm \|\cdot\|_\infty, admits an isometric embedding into the space \ell^\infty(K) of all bounded real-valued functions on K. This embedding is given by the evaluation map \phi: C(K) \to \ell^\infty(K) defined by (\phi f)(x) = f(x) for all x \in K, which preserves the uniform norm since \|\phi f\|_\infty = \sup_{x \in K} |f(x)| = \|f\|_\infty. This construction identifies C(K) as a closed subspace of \ell^\infty(K), highlighting the uniform norm's role in preserving the supremum metric structure across these spaces. A key property arising from this norm is that the uniform closure of any subset of C(K) consists entirely of continuous functions. Specifically, if a sequence of continuous functions \{f_n\} converges uniformly to a limit f in the \|\cdot\|_\infty norm, then f is continuous on K. This preservation of continuity under uniform limits underscores the completeness of C(K) as a Banach space and ensures that dense approximations remain within the space of continuous functions. The Weierstrass approximation theorem provides a foundational result on density in this setting: for a compact interval [a,b] \subset \mathbb{R}, the polynomials are dense in C[a,b] with respect to the uniform norm, meaning any continuous function on [a,b] can be uniformly approximated arbitrarily closely by polynomials.^[24] This theorem, originally established by Karl Weierstrass in 1885, demonstrates the uniform norm's utility in approximation theory by quantifying how well simple algebraic structures can approximate more complex continuous behaviors on bounded domains. The Stone-Weierstrass theorem extends this idea to more general compact Hausdorff spaces: if \mathcal{A} is a subalgebra of C(K) that contains the constants and separates points (i.e., for any distinct x,y \in K, there exists f \in \mathcal{A} with f(x) \neq f(y)), then \mathcal{A} is dense in C(K) under the uniform norm.^[25] Formulated by Marshall H. Stone in 1937 and refined in 1948, this result generalizes the Weierstrass theorem by showing that suitable algebras of continuous functions achieve uniform density, with applications in representing continuous functions via generating sets like trigonometric polynomials on circles or other structured subalgebras.

Applications in Analysis

In Functional Analysis and Banach Spaces

The space of all continuous real-valued functions on a compact Hausdorff topological space K, denoted C(K), equipped with the uniform norm \|f\|_\infty = \sup_{x \in K} |f(x)|, forms a Banach space.^[26] This completeness arises from the fact that uniform limits of continuous functions on compact sets are continuous, ensuring every Cauchy sequence converges in the norm.^[26] The dual space C(K)^* consists of all bounded linear functionals on C(K), which by the Riesz–Markov–Kakutani representation theorem, is isometrically isomorphic to the space of regular signed Borel measures on K.^[27] In the context of bounded linear operators on spaces equipped with the uniform norm, such as those between subspaces of C(K), the operator norm of a bounded linear operator T: X \to Y is defined as \|T\| = \sup \{ \|Tf\|_\infty / \|f\|_\infty : f \in X, f \neq 0 \}, or equivalently \|T\| = \sup \{ \|Tf\|_\infty : \|f\|_\infty \leq 1 \}.^[28] This norm measures the maximum uniform amplification of the input by T, and for operators preserving the uniform structure, it aligns directly with the supremum of the uniform norms over the unit ball.^[28] Boundedness of T is equivalent to continuity, a foundational property in the theory of operators on Banach spaces with uniform norms.^[26] The sequence space \ell^\infty(I) over an arbitrary index set I, consisting of all bounded functions from I to \mathbb{R} or \mathbb{C} with pointwise operations and the uniform norm, is a commutative unital Banach algebra.^[29] Its multiplication is pointwise, satisfying the algebra norm condition \|fg\|_\infty \leq \|f\|_\infty \|g\|_\infty, and completeness follows from the uniform norm's properties on bounded sequences.^[29] When I is uncountable, \ell^\infty(I) is non-separable, as it contains an uncountable discrete subset of characteristic functions with pairwise distance 1, preventing a countable dense subset.^[30] The space C[0,1] under the uniform norm fails to be reflexive, meaning the canonical embedding into its bidual C[0,1]^{**} is not surjective. The space C[0,1] under the uniform norm fails to be reflexive, as it contains a closed subspace isomorphic to the non-reflexive space c_0.^[31]

In Approximation Theory and Numerical Methods

In approximation theory, the uniform norm plays a central role in determining the best approximation of a continuous function f on a compact interval [a, b] by polynomials of degree at most n. The best uniform approximation is the unique polynomial p_n that minimizes \|f - p_n\|_\infty = \max_{x \in [a, b]} |f(x) - p_n(x)|. This minimizer is characterized by the equioscillation theorem, which asserts that the error function f - p_n attains its maximum absolute value at least n+2 points in [a, b], with the error alternating in sign at these points. This property holds more generally for approximations in Chebyshev systems, where the approximating subspace satisfies certain uniqueness conditions, ensuring the error equioscillates exactly n+2 times for the optimal approximation.^[32] Chebyshev polynomials exemplify the uniform norm's role in optimal approximation. The Chebyshev polynomial of the first kind, T_n(x), satisfies \|T_n\|_\infty = 1 on [-1, 1] and equioscillates n+1 times. Among all monic polynomials of degree n (leading coefficient 1), the scaled monic Chebyshev polynomial \hat{T}_n(x) = T_n(x)/2^{n-1} minimizes the uniform norm, achieving \|\hat{T}_n\|_\infty = 1/2^{n-1}. This minimal deviation property makes Chebyshev polynomials fundamental for constructing near-optimal approximations and understanding the growth of the best approximation error E_n(f) = \min_p \|f - p\|_\infty, which satisfies E_n(f) \leq (1 + \Lambda_n) \|f - p\|_\infty for any polynomial p of degree at most n, where \Lambda_n is the Lebesgue constant.^[32] The uniform norm also provides essential error bounds in polynomial interpolation. For Lagrange interpolation of f at n+1 points on [a, b], the error satisfies |f(x) - P_n(x)| = \frac{f^{(n+1)}(\xi)}{(n+1)!} \prod_{i=0}^n (x - x_i) for some \xi between the min and max of x, x_0, \dots, x_n. In the uniform norm, this yields \|f - P_n\|_\infty \leq \frac{\|f^{(n+1)}\|_\infty}{(n+1)!} \max_{x \in [a, b]} \left| \prod_{i=0}^n (x - x_i) \right|. For equidistant points with maximum spacing h, a bound is \|f - P_n\|_\infty \leq \frac{h^{n+1}}{4} \|f^{(n+1)}\|_\infty.^[33] This highlights the uniform norm's utility in quantifying worst-case interpolation accuracy, though equidistant points lead to exponential growth in the bound due to Runge's phenomenon.^[33] In numerical methods, the uniform norm measures maximum pointwise errors, ensuring reliability across the domain. For ordinary differential equation (ODE) solvers like Runge-Kutta methods applied to y' = f(t, y) on [t_0, T], the global error e(t) = y(t) - y_n(t) satisfies \|e\|_\infty \leq C h^p, where p is the method's order, h is the step size, and C depends on bounds involving \|f\| and Lipschitz constants in the uniform norm; this controls the maximum deviation over the interval. Similarly, in finite element methods for elliptic PDEs, maximum norm error estimates bound \|u - u_h\|_\infty \leq C h^{k} \|u^{(k+1)}\|_\infty for piecewise polynomials of degree k, providing pointwise guarantees crucial for applications requiring precise local accuracy, such as structural analysis. These estimates often rely on inverse inequalities and regularity assumptions to extend L^2-error bounds to the uniform norm.

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[PDF] AN INTRODUCTION TO NUMERICAL ANALYSIS Second Edition ...
Mar 2, 2012 · This introduction to numerical analysis was written for students in mathematics, the physical sciences, and engineering, ...