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Pointwise

In mathematics, pointwise is a qualifier used to describe properties, operations, or limits of functions that are evaluated or applied independently at each individual point in the domain of the function, rather than considering the functions as wholes or through aggregation over the domain.^[1]^[2] This approach contrasts with global or integral methods, such as those involving norms or convolutions, and is fundamental in analysis for defining structures on spaces of functions.^[1] Pointwise operations extend binary or unary operations from a codomain set, such as the real numbers, to functions by applying the operation to the function values at each point separately; for instance, the pointwise addition of two functions f and g from a set X to \mathbb{R} is defined as (f + g)(x) = f(x) + g(x) for all x \in X, and similarly for multiplication (f \cdot g)(x) = f(x) \cdot g(x) or scalar multiplication (\lambda f)(x) = \lambda f(x).^[1]^[3] These operations inherit algebraic properties like commutativity, associativity, and distributivity from the underlying set operations, making the set of functions a vector space or ring under pointwise definitions when applicable.^[1]^[3] Pointwise convergence, a key concept in real analysis, occurs when a sequence of functions \{f_k\} converges to a limit function f if, for every point x in the domain, the sequence of values f_k(x) converges to f(x) as k \to \infty; this is denoted f_k \to f pointwise and requires no measure on the domain.^[4]^[1] Unlike uniform convergence, which demands a uniform rate across the entire domain, pointwise convergence can fail to preserve important analytical properties like continuity or integrability of the limit.^[4] A related notion is pointwise almost everywhere convergence, which holds except on a set of measure zero, often used in measure theory contexts.^[4]

Basic Concepts

Definition of Pointwise Operation

In mathematics, a pointwise operation on functions refers to an operation that is performed independently at each point in the domain of the functions involved.^[1] More precisely, consider two functions f, g: D \to [R](/page/R), where D is a nonempty set serving as the common domain and R is a set equipped with a binary operation \oplus: R \times R \to R. The pointwise operation f \oplus g is then defined as the function from D to R given by (f \oplus g)(x) = f(x) \oplus g(x) for every x \in D.^[5]^[1] This construction extends naturally to n-ary pointwise operations: for functions f_1, \dots, f_n: D \to [R](/page/R) and an n-ary operation \oplus_n: R^n \to R, the pointwise result is the function (f_1 \oplus_n \cdots \oplus_n f_n): D \to [R](/page/R) satisfying (f_1 \oplus_n \cdots \oplus_n f_n)(x) = f_1(x) \oplus_n \cdots \oplus_n f_n(x) for all x \in D, provided the domain is the same for all functions and the codomain supports the operation.^[1] Scalar multiplication provides another instance, where for a scalar c \in [R](/page/R) (or from a compatible field) and function f: D \to [R](/page/R), the pointwise scalar multiple is (c \cdot f)(x) = c \cdot f(x) for each x \in D.^[1] In general, pointwise operations require mappings from a common domain to a codomain that admits the underlying operation, ensuring the result is well-defined point by point across the domain.^[5] Componentwise operations on finite-dimensional vectors represent a special case of pointwise operations when the domain is a finite discrete set.^[1]

Distinction from Other Operations

Pointwise operations on functions differ fundamentally from function composition, which combines functions by applying one to the output of the other across the entire domain. In function composition, defined as (f \circ g)(x) = f(g(x)), the value of the composed function at each point x depends on the intermediate evaluation g(x) as input to f, creating a global dependency that alters the input structure rather than operating directly on corresponding values.^[6] In contrast, pointwise operations, such as addition where (f + g)(x) = f(x) + g(x), treat each point independently without modifying the inputs, preserving the domain and enabling straightforward algebraic manipulation at each x. Unlike uniform operations, which assess functions holistically—such as in uniform convergence using the supremum norm to ensure consistent behavior across the domain—pointwise operations evaluate and combine values strictly at individual points without regard to global uniformity.^[7] For instance, uniform convergence requires that the maximum deviation between the sequence and its limit decreases uniformly over the domain, whereas pointwise operations, like pointwise limits, allow convergence rates to vary by location, potentially leading to disparate behaviors across different points.^[8] Pointwise notions often fail to align with their uniform counterparts, particularly in convergence, where a sequence may converge pointwise to a limit function but not uniformly, resulting in limits that do not preserve properties like continuity or integrability across the entire function.^[7] This discrepancy arises because pointwise evaluation ignores the supremum distance, permitting slower convergence near certain points without affecting the overall point-by-point agreement.^[8]

Pointwise Operations

Addition and Multiplication

Pointwise addition of two functions f and g, defined on the same domain X, is given by

([f + g](/page/F&G))(x) = f(x) + g(x)

for all x \in X.^[1] This operation is commonly applied in spaces such as the set of continuous functions C[a, b] on a closed interval [a, b], where the sum of two continuous functions remains continuous.^[9] The zero function, defined by z(x) = 0 for all x \in X, serves as the additive identity in such spaces.^[9] Pointwise multiplication of functions f and g is defined as (f \cdot g)(x) = f(x) \cdot g(x) for all x \in X, distinct from integral-based operations like convolution.^[1] Scalar multiplication, a special case, is (\lambda f)(x) = \lambda \cdot f(x) for a scalar \lambda in the codomain field, such as \mathbb{R}.^[1] Under pointwise addition and scalar multiplication, the set of functions from a set X to a vector space V (e.g., \mathbb{R}) forms a vector space, or more generally a module over the scalar field.^[9] With the inclusion of pointwise multiplication, when the codomain is a ring like \mathbb{R}, the set of all functions X \to \mathbb{R} forms a commutative ring.^[10] In the finite-dimensional case, such as functions from \{1, 2, \dots, n\} to \mathbb{R}, pointwise addition coincides with the standard vector addition in \mathbb{R}^n.^[11] This perspective generalizes to infinite domains, enabling the treatment of arbitrary function spaces as algebraic structures.^[10]

General Binary Operations

In mathematics, pointwise binary operations extend the concept of applying a binary operation defined on the codomain of functions to the functions themselves, performed independently at each point in the domain. Given two functions f, g: D \to C, where D is the domain and C is a set equipped with a binary operation \oplus: C \times C \to C, the pointwise operation is defined as (f \oplus g)(x) = f(x) \oplus g(x) for all x \in D, provided \oplus is defined for every pair of values in the image of f and g. This construction preserves the structure of the codomain algebra on the space of functions, such as vector spaces or lattices, and addition and multiplication serve as special cases when \oplus is respectively addition or multiplication in the codomain.^[12] A common example is the pointwise maximum operation in ordered sets, where ( \max(f, g) )(x) = \max( f(x), g(x) ), which arises naturally in the lattice structure of function spaces ordered pointwise. Similarly, pointwise exponentiation can be defined as (f^g)(x) = f(x)^{g(x)}, applicable when the codomain supports exponentiation, such as positive real numbers, ensuring the operation is well-defined across the domain. These operations highlight the versatility of pointwise definitions in constructing new functions from existing ones without altering the domain.^[13] For operations on sets, pointwise binary operations appear prominently with indicator (characteristic) functions, which map elements of a domain to \{0, 1\}. The pointwise union corresponds to the logical OR, so (\chi_{A \cup B})(x) = \chi_A(x) \lor \chi_B(x), where \chi_S(x) = 1 if x \in S and 0 otherwise, mirroring set union. Likewise, pointwise intersection uses logical AND: (\chi_{A \cap B})(x) = \chi_A(x) \land \chi_B(x). In lattice theory, these extend to the pointwise meet \wedge and join \vee on functions valued in a lattice, forming a function lattice under the pointwise order.^[13] In Boolean algebras, the set of all functions from a fixed domain to a Boolean algebra B, equipped with pointwise applications of the algebra's operations (such as AND, OR, and NOT), forms the function algebra, which itself is a Boolean algebra. This structure underlies Boolean functions in logic and computer science, where operations are computed pointwise to evaluate truth values across inputs. For matrix-valued functions, the Hadamard product exemplifies pointwise multiplication, defined entrywise as (A \circ B)_{ij} = a_{ij} b_{ij} for matrices A, B of compatible dimensions, preserving matrix properties like positivity under certain conditions.^[14]^[15] A key requirement for pointwise binary operations is domain compatibility: the operation \oplus must be defined on the codomain values f(x) and g(x) for every x \in D, which may impose restrictions such as restricting the domain to subsets where values are positive for exponentiation or bounded for max in incomplete orders. This ensures the resulting function is well-defined over the entire domain, enabling broad applications in analysis and algebra.^[12]

Pointwise Relations and Orders

Inequalities and Orders

In mathematics, particularly in order theory and functional analysis, the pointwise inequality between two functions f, g: D \to Y, where D is a domain and (Y, \leq) is a partially ordered set, is defined such that f \leq g if and only if f(x) \leq g(x) for every x \in D. This definition extends naturally to the other inequalities: f < g holds if f(x) < g(x) for all x \in D, while f \geq g if f(x) \geq g(x) for all x \in D, and f > g similarly. The pointwise order thus equips the set of all functions from D to Y with a relational structure derived directly from the order on Y.^[16] When (Y, \leq) is a poset, the pointwise order on the function space is itself a partial order. It is reflexive, as f \leq f follows from f(x) \leq f(x) for all x \in D; antisymmetric, since f \leq g and g \leq f imply f(x) = g(x) for all x \in D, hence f = g; and transitive, because if f \leq g and g \leq h, then f(x) \leq g(x) \leq h(x) for all x \in D, so f \leq h. These properties ensure that the pointwise order preserves the foundational structure of partial orders on the codomain, making it suitable for analyzing ordered function spaces. For instance, in the space of real-valued functions on D (where Y = \mathbb{R} with its standard order), the pointwise order f \leq g means f(x) \leq g(x) everywhere, forming a canonical example of a partially ordered set that often underlies lattice structures with pointwise minima and maxima.^[16] In finite-dimensional cases, such as when D is a finite set, the pointwise order coincides with the componentwise order on vectors in Y^{|D|}, where comparisons are made entry by entry. This discrete perspective highlights the pointwise order's role in embedding finite products of posets into a unified relational framework, though the general notation remains "pointwise" to emphasize independence from dimensionality.^[16]

Examples of Relations

A simple example of pointwise ordering arises with constant functions on a domain such as the real line. Consider the constant function f(x) = 1 for all x \in \mathbb{R} and the constant function g(x) = 2 for all x \in \mathbb{R}. Then f \leq g pointwise everywhere, since $1 \leq 2 holds for every x. This illustrates how pointwise relations reduce to the standard order on constants in the codomain, preserving the ordering in the function space. In probability theory, pointwise relations on cumulative distribution functions (CDFs) correspond to stochastic orders. Specifically, for two CDFs F and G on \mathbb{R}, F \leq G pointwise if and only if the associated random variables satisfy first-order stochastic dominance, meaning the distribution with CDF F is stochastically smaller than the one with CDF G. This equivalence provides a functional characterization of stochastic ordering, widely used in decision theory and risk analysis. Pointwise relations also induce lattice structures in spaces of bounded functions. In the space L^\infty(\mu) of essentially bounded measurable functions on a measure space (\Omega, \mu), the pointwise supremum \sup(f, g) and infimum \inf(f, g) (defined almost everywhere) exist for any f, g \in L^\infty(\mu), making L^\infty(\mu) a vector lattice under the pointwise order. For a bounded collection of functions, the pointwise essential supremum and infimum similarly form the lattice operations, enabling applications in optimization and approximation theory. A key preservation property of pointwise orders is their monotonicity with respect to integration. If f and g are nonnegative integrable functions on a measure space with f \leq g pointwise almost everywhere, then \int f \, d\mu \leq \int g \, d\mu. This follows from the monotonicity of the Lebesgue integral and holds without requiring additional dominated convergence assumptions.

Properties and Applications

Algebraic Properties

Pointwise operations on functions induce algebraic structures that mirror those of the codomain. For functions f, g: X \to \mathbb{R} from a non-empty set X to the reals, pointwise addition and scalar multiplication define a vector space structure, with properties such as commutativity and associativity inherited directly from the operations in \mathbb{R}. Specifically, addition is commutative because (f + g)(x) = f(x) + g(x) = g(x) + f(x) = (g + f)(x) for all x \in X, reflecting the commutativity of real addition.^[17] Associativity holds similarly: ((f + g) + h)(x) = (f(x) + g(x)) + h(x) = f(x) + (g(x) + h(x)) = (f + (g + h))(x), due to associativity in \mathbb{R}.^[17] Distributivity is preserved pointwise as well. Scalar multiplication over addition satisfies c(f + g)(x) = c(f(x) + g(x)) = c f(x) + c g(x) = (c f + c g)(x) for any scalar c \in \mathbb{R}, and addition of scalars over multiplication holds: (c + d) f(x) = (c + d) f(x) = c f(x) + d f(x) = ((c + d) f)(x).^[17] These properties ensure that the set of all such functions, often denoted \mathbb{R}^X, forms a vector space over \mathbb{R}. Restricting to continuous functions, the space C(X) of continuous real-valued functions on a topological space X inherits this vector space structure under the same pointwise operations, provided X is non-empty.^[18] The additive identity is the zero function, defined by $0(x) = 0 for all x \in X, satisfying f + 0 = f. Each function f has an additive inverse -f, where (-f)(x) = -f(x), ensuring f + (-f) = 0.^[17] Under pointwise multiplication, (f \cdot g)(x) = f(x) g(x), the space C(X) forms a commutative ring with unity when X is a topological space supporting continuous functions, as multiplication in \mathbb{R} is commutative and associative, inducing the same pointwise: f \cdot g = g \cdot f and (f \cdot g) \cdot h = f \cdot (g \cdot h). The multiplicative identity is the constant function $1(x) = 1.^[19]^[20]

Use in Analysis and Algebra

In functional analysis, pointwise convergence of a sequence of functions \{f_n\} to a function f on a domain X is defined by the condition that for every x \in X, f_n(x) \to f(x) as n \to \infty.^[4] This form of convergence is weaker than uniform convergence, as the rate of convergence may vary across points in X, and pointwise limits do not necessarily preserve important properties like continuity.^[8] For instance, the sequence f_n(x) = x^n on [0,1] converges pointwise to the discontinuous function f(x) = 0 for x \in [0,1) and f(1) = 1, despite each f_n being continuous.^[21] Pointwise convergence relates to norms through the supremum norm \|f\|_\infty = \sup_{x \in X} |f(x)|, which provides a global bound on pointwise values since |f(x)| \leq \|f\|_\infty for all x.^[22] In measure theory, Egorov's theorem strengthens pointwise almost everywhere convergence to almost uniform convergence on sets of finite measure, ensuring that for any \epsilon > 0, there exists a subset of measure less than \epsilon outside which the convergence is uniform.^[23] This result is pivotal for interchanging limits and integrals in L^p spaces.^[24] In algebra, pointwise operations arise naturally in the ring of functions from a set X to a ring R, equipped with pointwise addition and multiplication, forming a ring structure that generalizes scalar operations.^[10] Modules over such function rings involve pointwise actions, where scalar multiplication by a function \phi: X \to R acts componentwise on module elements indexed by X. For finite index sets, componentwise operations on modules like R^n coincide with pointwise definitions, enabling direct extensions to infinite cases in sheaf theory and algebraic geometry.^[25] The Hadamard product, or elementwise multiplication of matrices, exemplifies pointwise operations in multilinear algebra, where for compatible matrices A and B, the entry (A \circ B)_{ij} = a_{ij} b_{ij} preserves multilinear structures in tensor products and covariance analyses.^[26] This operation facilitates decompositions in multivariate settings, bridging pointwise control with global algebraic invariants.^[27]

References

[1]
pointwise - PlanetMath.org
Mar 22, 2013 · pointwise operations—operations defined on functions by applying the operations to function values separately for each point in the domain of ...
[2]
Definition of pointwise in mathematics? - Physics Forums
Jan 18, 2015 · The term "pointwise" in mathematics refers to operations on functions that are defined at each individual point in their domain.
[3]
[PDF] Section 1: (Binary) Operations It is assumed that you learned in Math ...
We know addition and multiplication of real numbers, addition of vectors and addition of ma- trices, pointwise addition and multiplication of functions all are ...
[4]
[PDF] 3.8 Three Types of Convergence - Christopher Heil
Recalling Definition 0.5, we say that fk converges pointwise to a function f if for each individual element x ∈ X, the scalar fk(x) converges to f(x) as k → ∞.
[5]
Definition:Pointwise Operation on Rational-Valued Functions
Jun 9, 2020 · Definition. Let S be a non-empty set. Let QS be the set of all mappings f:S→Q, where Q is the set of rational numbers.<|separator|>
[6]
6.1 Pointwise and uniform convergence
Pointwise convergence means a limit exists for each point, while uniform convergence means the limit exists for all points, and implies pointwise convergence.
[7]
[PDF] Chapter 9: Sequences and Series of Functions
Pointwise convergence defines the convergence of functions in terms of the conver- gence of their values at each point of their domain. Definition 9.1.
[8]
The development of the concept of uniform convergence in Karl ...
Dec 23, 2020 · The history of uniform convergence is typically focused on the contributions of Cauchy, Seidel, Stokes, and Björling. While the mathematical ...
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https://math.libretexts.org/Bookshelves/Linear_Algebra/Map%3A_Linear_Algebra_(Waldron_Cherney_and_Denton)/05%3A_Vector_Spaces/5.01%3A_Examples_of_Vector_Spaces
[10]
[PDF] Standard definitions for rings - Keith Conrad
Examples of rings are Z,. Q, all functions R → R with pointwise addition and multiplication, and M2(R) – the latter being a noncommutative ring – but 2Z is not ...
[11]
Addition of Vectors and Matrices
If we do, the pointwise function addition will be reduced to the componentwise addition of vectors. Notations reinforce the analogy. The space of n-\mbox{tuples} ...
[12]
[PDF] Companion to Functional Analysis John M. Erdman
May 9, 2022 · In a functional analysis course the focus is shifted to spaces of such functions ... under pointwise operations of addition and scalar ...
[13]
[PDF] Chapter 5. Lattices, closure operators, and Galois connections.
The set of real-valued functions on any set X may be ordered by writing f ≤ g if f(x) ≤ g(x) for all x, and this set is a lattice under pointwise maximum and ...
[14]
[PDF] Boolean Equations
Apr 3, 2025 · For any Boolean functions f : Bn → B,g : Bn → B the functions a) f ∪ g : Bn → B : x 7→ f(x) ∪ g(x) b) f ∩ g : Bn → B : x 7→ f(x) ∩ g(x) c) f : ...
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The Hadamard product (Chapter 5) - Topics in Matrix Analysis
>Topics in Matrix Analysis; >The Hadamard product. You have Access. 14: Cited by. Chapter 5 - The Hadamard product. Published online by Cambridge University ...
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None
Below is a merged summary of "Pointwise Order" in Garrett Birkhoff's *Lattice Theory* (1948, Revised Edition), consolidating all the information from the provided segments into a single, comprehensive response. To maximize detail and clarity, I will use a table in CSV format to organize the key aspects (Definition, Context, Properties, and Useful URLs) across the relevant sections of the book. Following the table, I will provide a narrative summary to tie everything together and address any gaps or overlaps.
[17]
[PDF] 18.102 S2021 Lecture 1. Basic Banach Space Theory
n and Cn are vector spaces, and so is C([0, 1]), the space of continuous functions [0, 1] → C. (This last example is indeed a vector space because the sum ...
[18]
[PDF] 10. Euclidean Spaces
Sep 13, 2022 · The space of continuous functions on A is ... This space is a vector space when vector addition and scalar multiplication are defined pointwise.<|separator|>
[19]
semiclean rings and rings of continuous functions - Project Euclid
Sep 28, 2012 · the set of all continuous A-valued functions on X. Under pointwise addition and multiplication C(X, A) is a commutative ring with unity.
[20]
[PDF] Ring of continuous functions - Universidad de Granada
The set C(X) has a ring structure: the sum and the product are defined pointwise, and the constant function equal to 1 plays the role of the unity. The main ...<|separator|>
[21]
[PDF] Math 410 Section 9.2: Pointwise Convergence of Functions
Example, Destroying Continuity: Consider fn : [0, 1] → R given by f(x) ... It follows that pointwise convergence does not preserve continuity. Page 3. 5 ...
[22]
[PDF] Functional Analysis - Purdue Math
where the norm of the derivatives is the sup norm. Proof. It is straightforward to verify that Ck([a, b]) is a vector space and that Norm is a norm.
[23]
[PDF] 3.9 Egoroff's Theorem - Christopher Heil
We know that pointwise convergence of functions does not imply uniform con- vergence, and likewise pointwise a.e. convergence does not imply L∞ norm convergence ...
[24]
[PDF] LECTURE 13 Egoroff 's theorem (pointwise convergence is nearly ...
Egoroff 's theorem (pointwise convergence is nearly uniform. Theorem 0.1. Suppose µ(X) < ∞. Let fn : X C be a sequence of → measurable functions such that fn f ...
[25]
[PDF] some basic module-theoretic notions and examples
the most natural example is the ring C∞(R) of smooth functions R → R with pointwise operations. The set of functions in C∞(R) whose derivatives of all ...
[26]
Hadamard products and multivariate statistical analysis
The Hadamard product of two matrices multiplied together elementwise is a rather neglected concept in matrix theory and has found only brief and scattered ...Missing: multilinear | Show results with:multilinear
[27]
Hadamard product in nLab
Aug 2, 2023 · Definition. Of matrices. The Hadamard product or Schur product or element-wise product of two matrices A A and B B in a commutative ring R R ...