Hubbry Logo
PointwisePointwiseMain
Open search
Pointwise
Community hub
Pointwise
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Pointwise
Pointwise
Pointwise
View on Wikipedia
from Wikipedia

In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value of some function An important class of pointwise concepts are the pointwise operations, that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition. Important relations can also be defined pointwise.

Pointwise operations

[edit]
Pointwise sum (upper plot, violet) and product (green) of the functions sin (lower plot, blue) and ln (red). The highlighted vertical slice shows the computation at the point x=2π.

Formal definition

[edit]

A binary operation o: Y × Y → Y on a set Y can be lifted pointwise to an operation O: (X→Y) × (X→Y) → (X→Y) on the set X → Y of all functions from X to Y as follows: Given two functions f1: X → Y and f2: X → Y, define the function O(f1, f2): X → Y by

(O(f1, f2))(x) = o(f1(x), f2(x)) for all x ∈ X.

Commonly, o and O are denoted by the same symbol. A similar definition is used for unary operations o, and for operations of other arity.[citation needed]

Examples

[edit]

The pointwise addition of two functions and with the same domain and codomain is defined by:

The pointwise product or pointwise multiplication is:

The pointwise product with a scalar is usually written with the scalar term first. Thus, when is a scalar:

An example of an operation on functions which is not pointwise is convolution.

Properties

[edit]

Pointwise operations inherit such properties as associativity, commutativity and distributivity from corresponding operations on the codomain. If is some algebraic structure, the set of all functions to the carrier set of can be turned into an algebraic structure of the same type in an analogous way.

Componentwise operations

[edit]

Componentwise operations are usually defined on vectors, where vectors are elements of the set for some natural number and some field . If we denote the -th component of any vector as , then componentwise addition is .

Componentwise operations can be defined on matrices. Matrix addition, where is a componentwise operation while matrix multiplication is not.

A tuple can be regarded as a function, and a vector is a tuple. Therefore, any vector corresponds to the function such that , and any componentwise operation on vectors is the pointwise operation on functions corresponding to those vectors.

Pointwise relations

[edit]

In order theory it is common to define a pointwise partial order on functions. With A, B posets, the set of functions A → B can be ordered by defining f ≤ g if (∀x ∈ A) f(x) ≤ g(x). Pointwise orders also inherit some properties of the underlying posets. For instance if A and B are continuous lattices, then so is the set of functions A → B with pointwise order.[1] Using the pointwise order on functions one can concisely define other important notions, for instance:[2]

An example of an infinitary pointwise relation is pointwise convergence of functions—a sequence of functions with converges pointwise to a function f if for each x in X

Notes

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Pointwise

View on Grokipedia
from Grokipedia
In , 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. This approach contrasts with global or methods, such as those involving norms or convolutions, and is fundamental in for defining structures on spaces of functions. Pointwise operations extend binary or unary operations from a set, such as the real numbers, to functions by applying the operation to the function values at each point separately; for instance, the pointwise of two functions ff and gg from a set XX to R\mathbb{R} is defined as (f+g)(x)=f(x)+g(x)(f + g)(x) = f(x) + g(x) for all x∈Xx \in X, and similarly for (f⋅g)(x)=f(x)⋅g(x)(f \cdot g)(x) = f(x) \cdot g(x) or (λf)(x)=λf(x)(\lambda f)(x) = \lambda f(x). These operations inherit algebraic properties like commutativity, associativity, and distributivity from the underlying set operations, making the set of functions a or ring under pointwise definitions when applicable. , a key concept in , occurs when a of functions {fk}\{f_k\} converges to a limit function ff if, for every point xx in the domain, the of values fk(x)f_k(x) converges to f(x)f(x) as k→∞k \to \infty; this is denoted fk→ff_k \to f pointwise and requires no measure on the domain. Unlike , 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. A related notion is pointwise almost everywhere convergence, which holds except on a set of measure zero, often used in measure theory contexts.

Basic Concepts

Definition of Pointwise Operation

In , a pointwise operation on functions refers to an operation that is performed independently at each point in the domain of the functions involved. More precisely, consider two functions f,g:D→[R](/page/R)f, g: D \to [R](/page/R), where DD is a nonempty set serving as the common domain and RR is a set equipped with a ⊕:R×R→R\oplus: R \times R \to R. The pointwise operation f⊕gf \oplus g is then defined as the function from DD to RR given by (f⊕g)(x)=f(x)⊕g(x)(f \oplus g)(x) = f(x) \oplus g(x) for every x∈Dx \in D. This construction extends naturally to n-ary pointwise operations: for functions f1,…,fn:D→[R](/page/R)f_1, \dots, f_n: D \to [R](/page/R) and an n-ary operation ⊕n:Rn→R\oplus_n: R^n \to R, the pointwise result is the function (f1⊕n⋯⊕nfn):D→[R](/page/R)(f_1 \oplus_n \cdots \oplus_n f_n): D \to [R](/page/R) satisfying (f1⊕n⋯⊕nfn)(x)=f1(x)⊕n⋯⊕nfn(x)(f_1 \oplus_n \cdots \oplus_n f_n)(x) = f_1(x) \oplus_n \cdots \oplus_n f_n(x) for all x∈Dx \in D, provided the domain is the same for all functions and the codomain supports the operation. Scalar multiplication provides another instance, where for a scalar c∈[R](/page/R)c \in [R](/page/R) (or from a compatible field) and function f:D→[R](/page/R)f: D \to [R](/page/R), the pointwise scalar multiple is (c⋅f)(x)=c⋅f(x)(c \cdot f)(x) = c \cdot f(x) for each x∈Dx \in D. In general, pointwise operations require mappings from a common domain to a that admits the underlying operation, ensuring the result is well-defined point by point across the domain. Componentwise operations on finite-dimensional vectors represent a special case of pointwise operations when the domain is a finite discrete set.

Distinction from Other Operations

Pointwise operations on functions differ fundamentally from , which combines functions by applying one to the output of the other across the entire domain. In , defined as (f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x)), the value of the composed function at each point xx depends on the intermediate evaluation g(x)g(x) as input to ff, creating a global dependency that alters the input structure rather than operating directly on corresponding values. In contrast, pointwise operations, such as where (f+g)(x)=f(x)+g(x)(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 xx. Unlike uniform operations, which assess functions holistically—such as in 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. 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. Pointwise notions often fail to align with their 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 . This discrepancy arises because pointwise evaluation ignores the supremum distance, permitting slower convergence near certain points without affecting the overall point-by-point agreement.

Pointwise Operations

Addition and Multiplication

Pointwise addition of two functions ff and gg, defined on the same domain XX, is given by ([f + g](/page/F&G))(x) = f(x) + g(x) for all x∈Xx \in X. This operation is commonly applied in spaces such as the set of continuous functions C[a,b]C[a, b] on a closed interval [a,b][a, b], where the sum of two continuous functions remains continuous. The zero function, defined by z(x)=0z(x) = 0 for all x∈Xx \in X, serves as the in such spaces. Pointwise multiplication of functions ff and gg is defined as (f⋅g)(x)=f(x)⋅g(x)(f \cdot g)(x) = f(x) \cdot g(x) for all x∈Xx \in X, distinct from integral-based operations like . , a special case, is (λf)(x)=λ⋅f(x)(\lambda f)(x) = \lambda \cdot f(x) for a scalar λ\lambda in the field, such as R\mathbb{R}. Under pointwise addition and , the set of functions from a set XX to a VV (e.g., R\mathbb{R}) forms a , or more generally a module over the . With the inclusion of pointwise multiplication, when the is a ring like R\mathbb{R}, the set of all functions X→RX \to \mathbb{R} forms a . In the finite-dimensional case, such as functions from {1,2,…,n}\{1, 2, \dots, n\} to R\mathbb{R}, pointwise addition coincides with the standard vector addition in Rn\mathbb{R}^n. This perspective generalizes to infinite domains, enabling the treatment of arbitrary function spaces as algebraic structures.

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→Cf, g: D \to C, where DD is the domain and CC is a set equipped with a binary operation ⊕:C×C→C\oplus: C \times C \to C, the pointwise operation is defined as (f⊕g)(x)=f(x)⊕g(x)(f \oplus g)(x) = f(x) \oplus g(x) for all x∈Dx \in D, provided ⊕\oplus is defined for every pair of values in the image of ff and gg. 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. A common example is the pointwise maximum operation in ordered sets, where (max⁡(f,g))(x)=max⁡(f(x),g(x))( \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 (fg)(x)=f(x)g(x)(f^g)(x) = f(x)^{g(x)}, applicable when the codomain supports , such as , 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. For operations on sets, pointwise binary operations appear prominently with indicator (characteristic) functions, which map elements of a domain to {0,1}\{0, 1\}. The pointwise union corresponds to the logical OR, so (χA∪B)(x)=χA(x)∨χB(x)(\chi_{A \cup B})(x) = \chi_A(x) \lor \chi_B(x), where χS(x)=1\chi_S(x) = 1 if x∈Sx \in S and 0 otherwise, mirroring set union. Likewise, pointwise uses logical AND: (χA∩B)(x)=χA(x)∧χB(x)(\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. In , the set of all functions from a fixed domain to a Boolean algebra BB, 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 , where operations are computed pointwise to evaluate truth values across inputs. For matrix-valued functions, the Hadamard product exemplifies pointwise , defined entrywise as (A∘B)ij=aijbij(A \circ B)_{ij} = a_{ij} b_{ij} for matrices A,BA, B of compatible dimensions, preserving matrix properties like positivity under certain conditions. A key requirement for pointwise binary operations is domain compatibility: the operation ⊕\oplus must be defined on the codomain values f(x)f(x) and g(x)g(x) for every x∈Dx \in D, which may impose restrictions such as restricting the domain to subsets where values are positive for or bounded for max in incomplete orders. This ensures the resulting function is well-defined over the entire domain, enabling broad applications in and .

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→Yf, g: D \to Y, where DD is a domain and (Y,≤)(Y, \leq) is a partially ordered set, is defined such that f≤gf \leq g if and only if f(x)≤g(x)f(x) \leq g(x) for every x∈Dx \in D. This definition extends naturally to the other inequalities: f<gf < g holds if f(x)<g(x)f(x) < g(x) for all x∈Dx \in D, while f≥gf \geq g if f(x)≥g(x)f(x) \geq g(x) for all x∈Dx \in D, and f>gf > g similarly. The pointwise order thus equips the set of all functions from DD to YY with a relational structure derived directly from the order on YY. When (Y,≤)(Y, \leq) is a poset, the pointwise order on the is itself a partial order. It is reflexive, as f≤ff \leq f follows from f(x)≤f(x)f(x) \leq f(x) for all x∈Dx \in D; antisymmetric, since f≤gf \leq g and g≤fg \leq f imply f(x)=g(x)f(x) = g(x) for all x∈Dx \in D, hence f=gf = g; and transitive, because if f≤gf \leq g and g≤hg \leq h, then f(x)≤g(x)≤h(x)f(x) \leq g(x) \leq h(x) for all x∈Dx \in D, so f≤hf \leq h. These properties ensure that the pointwise order preserves the foundational structure of partial orders on the , making it suitable for analyzing ordered function spaces. For instance, in the space of real-valued functions on DD (where Y=RY = \mathbb{R} with its standard order), the pointwise order f≤gf \leq g means f(x)≤g(x)f(x) \leq g(x) everywhere, forming a example of a that often underlies lattice structures with pointwise minima and maxima. In finite-dimensional cases, such as when DD is a , the pointwise order coincides with the componentwise order on vectors in Y∣D∣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.

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)=1f(x) = 1 for all x∈Rx \in \mathbb{R} and the constant function g(x)=2g(x) = 2 for all x∈Rx \in \mathbb{R}. Then f≤gf \leq g pointwise everywhere, since 1≤21 \leq 2 holds for every xx. This illustrates how pointwise relations reduce to the standard order on constants in the , preserving the ordering in the . In , pointwise relations on cumulative distribution functions (CDFs) correspond to stochastic orders. Specifically, for two CDFs FF and GG on R\mathbb{R}, F≤GF \leq G pointwise the associated random variables satisfy first-order , meaning the distribution with CDF FF is stochastically smaller than the one with CDF GG. This equivalence provides a functional characterization of stochastic ordering, widely used in and risk analysis. Pointwise relations also induce lattice structures in spaces of bounded functions. In the space L∞(μ)L^\infty(\mu) of essentially bounded measurable functions on a measure space (Ω,μ)(\Omega, \mu), the pointwise supremum sup⁡(f,g)\sup(f, g) and infimum inf⁡(f,g)\inf(f, g) (defined almost everywhere) exist for any f,g∈L∞(μ)f, g \in L^\infty(\mu), making L∞(μ)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, applications in optimization and . A key preservation property of pointwise orders is their monotonicity with respect to integration. If ff and gg are nonnegative integrable functions on a with f≤gf \leq g pointwise , then ∫f dμ≤∫g dμ\int f \, d\mu \leq \int g \, d\mu. This follows from the monotonicity of the Lebesgue 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 . For functions f,g:X→Rf, g: X \to \mathbb{R} from a non-empty set XX to the reals, pointwise and define a structure, with properties such as commutativity and associativity inherited directly from the operations in R\mathbb{R}. Specifically, is commutative because (f+g)(x)=f(x)+g(x)=g(x)+f(x)=(g+f)(x)(f + g)(x) = f(x) + g(x) = g(x) + f(x) = (g + f)(x) for all x∈Xx \in X, reflecting the commutativity of real . Associativity holds similarly: ((f+g)+h)(x)=(f(x)+g(x))+h(x)=f(x)+(g(x)+h(x))=(f+(g+h))(x)((f + g) + h)(x) = (f(x) + g(x)) + h(x) = f(x) + (g(x) + h(x)) = (f + (g + h))(x), due to associativity in R\mathbb{R}. Distributivity is preserved pointwise as well. Scalar multiplication over addition satisfies c(f+g)(x)=c(f(x)+g(x))=cf(x)+cg(x)=(cf+cg)(x)c(f + g)(x) = c(f(x) + g(x)) = c f(x) + c g(x) = (c f + c g)(x) for any scalar c∈Rc \in \mathbb{R}, and addition of scalars over multiplication holds: (c+d)f(x)=(c+d)f(x)=cf(x)+df(x)=((c+d)f)(x)(c + d) f(x) = (c + d) f(x) = c f(x) + d f(x) = ((c + d) f)(x). These properties ensure that the set of all such functions, often denoted RX\mathbb{R}^X, forms a over R\mathbb{R}. Restricting to continuous functions, the space C(X)C(X) of continuous real-valued functions on a XX inherits this vector space structure under the same pointwise operations, provided XX is non-empty. The additive identity is the zero function, defined by 0(x)=00(x) = 0 for all x∈Xx \in X, satisfying f+0=ff + 0 = f. Each function ff has an additive inverse −f-f, where (−f)(x)=−f(x)(-f)(x) = -f(x), ensuring f+(−f)=0f + (-f) = 0. Under pointwise multiplication, (f⋅g)(x)=f(x)g(x)(f \cdot g)(x) = f(x) g(x), the space C(X)C(X) forms a commutative ring with unity when XX is a topological space supporting continuous functions, as multiplication in R\mathbb{R} is commutative and associative, inducing the same pointwise: f⋅g=g⋅ff \cdot g = g \cdot f and (f⋅g)⋅h=f⋅(g⋅h)(f \cdot g) \cdot h = f \cdot (g \cdot h). The multiplicative identity is the constant function 1(x)=11(x) = 1.

Use in Analysis and Algebra

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

References

  1. https://planetmath.org/pointwise
  2. https://www.physicsforums.com/threads/definition-of-pointwise-in-mathematics.792994/
  3. https://math.colgate.edu/math320/dlantz/extras/notes01.pdf
  4. https://heil.math.gatech.edu/6337/spring11/section3.8.pdf
  5. https://proofwiki.org/wiki/Definition:Pointwise_Operation_on_Rational-Valued_Functions
  6. https://mathworld.wolfram.com/Composition.html
  7. https://www.jirka.org/ra/html/sec_puconv.html
  8. https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch9.pdf
  9. 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. https://kconrad.math.uconn.edu/blurbs/ringtheory/ringdefs.pdf
  11. https://www.cut-the-knot.org/do_you_know/add_vec.shtml
  12. https://web.pdx.edu/~erdman/CFA/functional_analysis_pdf.pdf
  13. https://math.berkeley.edu/~gbergman/245/2.4/Ch.5.pdf
  14. https://dmg.tuwien.ac.at/hetzl/teaching/s_gruber_ws24.pdf
  15. https://www.cambridge.org/core/books/topics-in-matrix-analysis/hadamard-product/1614DD81C7ED99043B97523198550A63
  16. https://math.chapman.edu/~jipsen/summerschool/Birkhoff%201948%20Lattice%20Theory%20Revised%20Edition.pdf
  17. https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/4ae9a5af8622dba3aa37461b71d08407_MIT18_102s21_lec1.pdf
  18. https://faculty.fiu.edu/~boydj/mathecon/math10.pdf
  19. https://projecteuclid.org/journals/journal-of-commutative-algebra/volume-6/issue-1/Semiclean-rings-and-rings-of-continuous-functions/10.1216/JCA-2014-6-1-1.pdf
  20. http://www.ugr.es/~anillos/textos/pdf/2023/TFG_Continua.pdf
  21. https://math.umd.edu/~immortal/MATH410/lecturenotes/ch9-2.pdf
  22. https://www.math.purdue.edu/~iswanso/functionalanalysis.pdf
  23. https://heil.math.gatech.edu/6337/spring11/section3.9.pdf
  24. https://ocw.mit.edu/courses/18-125-measure-and-integration-fall-2003/8ee188461695d9db958de0347471d0db_18125_lec13.pdf
  25. http://math.stanford.edu/~conrad/154Page/handouts/moduleintro.pdf
  26. https://www.sciencedirect.com/science/article/pii/0024379573900232
  27. https://ncatlab.org/nlab/show/Hadamard%2Bproduct
Add your contribution
Related Hubs
User Avatar
No comments yet.