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| In [[mathematics]], the qualifier '''pointwise''' is used to indicate that a certain property is defined by considering each value <math>f(x)</math> of some function <math>f.</math> An important class of pointwise concepts are the ''pointwise operations'' — operations defined on functions by applying the operations to function values separately for each point in the [[domain (mathematics)|domain]] of definition. Important [[Theory of relations|relations]] can also be defined pointwise.
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| == Pointwise operations ==
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| Examples include
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| : <math>
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| \begin{align}
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| (f+g)(x) & = f(x)+g(x) & \text{(pointwise addition)} \\
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| (f\cdot g)(x) & = f(x) \cdot g(x) & \text{(pointwise multiplication)} \\
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| (\lambda f)(x) & = \lambda \cdot f(x) & \text{(pointwise multiplication by a scalar)}
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| \end{align}
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| </math>
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| where <math>f,g:X\to R</math>.
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| See [[pointwise product]], [[scalar (mathematics)|scalar]].
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| Pointwise operations inherit such properties as [[associativity]], [[commutativity]] and [[distributivity]] from corresponding operations on the [[codomain]]. An example of an operation on functions which is ''not'' pointwise is [[convolution]].
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| By taking some [[algebraic structure]] <math>A</math> in the place of <math>R</math>, we can turn the set of all functions <math>X</math> to the [[carrier set]] of <math>A</math> into an algebraic structure of the same type in an analogous way.
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| == Componentwise operations ==
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| Componentwise operations are usually defined on vectors, where vectors are elements of the set <math>K^n</math> for some [[natural number]] <math>n</math> and some [[Field (mathematics)|field]] <math>K</math>. <math>K</math> can be generalized to a set. If we denote the <math>i</math>-th component of any vector <math>v</math> as <math>v_i</math>, then componentwise addition is <math>(u+v)_i = u_i+v_i</math>.
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| A [[Tuple#Tuples_as_functions|tuple]] can be regarded as a function, and a vector is a tuple. Therefore any vector <math>v</math> corresponds to the function <math>f:n\to K</math> such that <math>f(i)=v_i</math>, and any componentwise operation on vectors is the pointwise operation on functions corresponding to those vectors.
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| == Pointwise relations ==
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| In [[order theory]] it is common to define a pointwise [[partial order]] on functions. With ''A'', ''B'' [[partially ordered sets|posets]], the set of functions ''A'' → ''B'' can be ordered by ''f'' ≤ ''g'' if and only 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 lattice]]s, then so is the set of functions ''A'' → ''B'' with pointwise order.<ref>Gierz, p. xxxiii</ref> Using the pointwise order on functions one can concisely define other important notions, for instance:<ref>Gierz, p. 26</ref>
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| * A [[closure operator]] ''c'' on a poset ''P'' is a [[monotonic function|monotone]] and [[idempotent]] self-map on ''P'' (i.e. a [[projection (order)|projection operator]]) with the additional property that id<sub>''A''</sub> ≤ ''c'', where id is the [[identity function]].
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| * Similarly, a projection operator ''k'' is called a [[kernel operator]] if and only if ''k'' ≤ id<sub>''A''</sub>.
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| An example of [[infinitary]] pointwise relation is [[pointwise convergence]] of functions — a [[sequence]] of functions
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| :<math>\{f_n\}_{n=1}^\infty</math>
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| with
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| :<math>f_n:X \longrightarrow Y</math>
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| [[limit of a sequence|converges]] pointwise to a function <math>f</math> if for each <math>x</math> in <math>X</math>
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| :<math>\lim_{n \rightarrow \infty} f_n(x) = f(x).</math>
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| == Notes ==
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| {{reflist}}
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| == References ==
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| ''For order theory examples:''
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| * T.S. Blyth, ''Lattices and Ordered Algebraic Structures'', Springer, 2005, ISBN 1-85233-905-5.
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| * G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott: ''Continuous Lattices and Domains'', Cambridge University Press, 2003.
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| {{PlanetMath attribution|id=7260|title=Pointwise}}
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| [[Category:Mathematical terminology]]
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