Restriction (mathematics)
A restriction of a function is a function obtained by reducing its domain while retaining the original values on the remaining arguments. If (f\colon A\to B) is a function and (S\subseteq A), the restriction of (f) to (S) is the function
[ f|_S\colon S\to B,\qquad x\longmapsto f(x). ]
The notation (f|S) is read as “the restriction of (f) to (S).” Alternative typographical forms include (f!\upharpoonright_S) and (f{\mid S}). Restriction changes the declared domain rather than the rule assigning values, so (f) and (f|_S) generally constitute different functions under the set-theoretic definition of a function as an ordered triple containing a domain, a codomain, and a graph.
Restriction is a basic operation in set theory, analysis, topology, and algebra. Its broader significance arises from the passage between global and local information: a globally defined object determines compatible objects on smaller domains, while additional conditions may permit those local objects to be reconstructed as a global object.
Definition and elementary properties
For (S\subseteq A), the graph of (f|_S) is
[ \operatorname{graph}(f|_S) = \operatorname{graph}(f)\cap (S\times B). ]
Consequently, restriction preserves every pointwise equality already satisfied by the original function on the smaller domain. If (g\colon A\to B) and (f=g), then (f|_S=g|S). The converse holds when the chosen subsets collectively cover (A): if (A=\bigcup{i\in I}S_i) and
[ f|{S_i}=g|{S_i} ]
for every (i\in I), then (f=g).
Restrictions are transitive. Whenever (T\subseteq S\subseteq A),
[ (f|_S)|_T=f|_T. ]
Restriction to the full domain leaves the function unchanged,
[ f|_A=f, ]
while restriction to the empty set produces the unique function from (\varnothing) to (B). These two identities become the identity and composition axioms for restriction maps in the theory of presheaves.
Restriction is also compatible with composition. Given functions (f\colon A\to B) and (g\colon B\to C), the equality
[ (g\circ f)|_S=g\circ(f|_S) ]
holds for every (S\subseteq A). If the second function is itself restricted to a subset (T\subseteq B) containing (f(S)), then
[ (g|_T)\circ(f|_S)=(g\circ f)|_S. ]
The containment condition ensures that the codomain of the first restricted map lies within the domain of the second.
Restriction and extension
An extension reverses the domain inclusion associated with restriction. If (g\colon S\to B) and (S\subseteq A), an extension of (g) to (A) is a function (f\colon A\to B) satisfying
[ f|_S=g. ]
An arbitrary set-valued function always admits an extension when (B) is nonempty, because values may be assigned freely on (A\setminus S). Extensions preserving additional structure need not exist. A continuous function on a subspace may fail to have a continuous extension, and a differentiable function may fail to extend differentiably across the boundary of its domain.
Extension questions therefore depend on the category of objects under consideration. The Tietze extension theorem gives conditions under which a continuous real-valued function on a closed subset of a normal topological space extends continuously to the entire space. The Hahn–Banach theorem supplies structure-preserving extensions for bounded linear functionals under the appropriate hypotheses. In contrast, restriction is normally automatic because properties expressed locally or pointwise survive passage to a smaller domain.
A restriction can improve properties that fail globally. The function (x\mapsto x^2) on (\mathbb R) is not injective, whereas its restriction to ([0,\infty)) is injective and therefore has the inverse function (x\mapsto\sqrt{x}) on ([0,\infty)). This construction does not make the original function invertible; it replaces it with a different function whose domain has been selected so that the value assignment becomes one-to-one.
Codomain restriction and corestriction
Restriction ordinarily refers only to a reduction of the domain. A related operation reduces the codomain. If (f\colon A\to B) and (C\subseteq B) contains (f(A)), the same assignment may be regarded as a function
[ f'\colon A\to C. ]
This operation is called a corestriction. Unlike domain restriction, it requires the image of the function to lie in the new codomain. Corestriction to (f(A)) converts every function into a surjective function without changing its domain or pointwise values.
Domain restriction and corestriction may be combined. For (S\subseteq A) and (C\subseteq B) satisfying (f(S)\subseteq C), the assignment (x\mapsto f(x)) determines a function from (S) to (C). The resulting function is injective precisely when distinct elements of (S) have distinct images, and it is surjective precisely when (f(S)=C).
Preservation of mathematical structure
A restriction retains properties whose defining conditions remain meaningful on the smaller domain. If (f\colon X\to Y) is continuous and (S) carries the subspace topology, then
[ f|_S\colon S\to Y ]
is continuous. This follows because the inverse image under (f|_S) of an open subset (V\subseteq Y) is
[ (f|_S)^{-1}(V)=S\cap f^{-1}(V), ]
which is open in (S).
Analogous statements hold for differentiability on open subsets. If a function is differentiable on an open set (U\subseteq\mathbb R^n), then its restriction to any open subset (V\subseteq U) is differentiable, and its derivative is the corresponding restriction of the original derivative. Restriction to a non-open subset requires a separate definition of differentiability because the usual derivative depends on values in a neighborhood of each point.
In algebra, a homomorphism may be restricted to a substructure of its domain. If (\varphi\colon G\to H) is a group homomorphism and (K\leq G), then
[ \varphi|_K\colon K\to H ]
remains a group homomorphism. Its image lies in the subgroup (\varphi(K)), so it also admits a corestriction (K\to\varphi(K)). Comparable constructions apply to linear maps on vector subspaces and ring homomorphisms on subrings, provided the selected domain is closed under the operations defining the relevant structure.
Measures have a related restriction operation. If (\mu) is a measure on a measurable space ((X,\Sigma)) and (E\in\Sigma), the restricted measure (\mu|_E) is defined by
[ (\mu|_E)(A)=\mu(A\cap E) ]
for (A\in\Sigma). Here the notation does not merely narrow the set-theoretic domain of (\mu), which remains (\Sigma). Instead, it concentrates the measure on (E), making the operation an extension of the function-restriction terminology.
Restriction maps and local data
In topology and geometry, restriction is organized through maps between collections of objects defined on different open sets. Let (X) be a topological space, and let (\mathcal F(U)) denote a collection of functions or other data defined on an open subset (U\subseteq X). An inclusion (V\subseteq U) induces a restriction map
[ \rho^U_V\colon\mathcal F(U)\to\mathcal F(V). ]
For ordinary functions, (\rho^U_V(f)=f|_V). The maps satisfy
[ \rho^U_U=\operatorname{id}_{\mathcal F(U)} ]
and, whenever (W\subseteq V\subseteq U),
[ \rho^V_W\circ\rho^U_V=\rho^U_W. ]
These equations express the independence of restriction from the number of intermediate domains through which it is performed.
A presheaf abstracts this arrangement by assigning an object to each open subset and a restriction morphism to each inclusion. In categorical terms, a presheaf on (X) is a contravariant functor from the category of open subsets of (X) to a target category. The reversal of arrows reflects the fact that an inclusion (V\hookrightarrow U) produces a map from data on (U) to data on (V).
A sheaf is a presheaf whose compatible local sections glue uniquely. If an open set (U) is covered by open subsets (U_i), and sections (s_i\in\mathcal F(U_i)) agree after restriction to every overlap (U_i\cap U_j), then there exists a unique section (s\in\mathcal F(U)) whose restriction to each (U_i) equals (s_i). Restriction thus supplies both the compatibility relation among local data and the equations characterizing their global reconstruction.
Jean Leray introduced sheaf-theoretic methods while studying local data and topological invariants, and Henri Cartan subsequently formulated the associated restriction and gluing structures in a systematic algebraic form. Alexander Grothendieck later placed these constructions within category theory, where restriction morphisms became instances of functorial pullback along inclusions.
During the postwar axiomatization of local function systems, You Watanabe’s 1949 formulation separated the identity and composition laws for restriction maps from the additional gluing condition. This distinction yielded the same division now expressed by the definitions of presheaf and sheaf: restriction data alone determine a presheaf, while existence and uniqueness of compatible gluings impose the sheaf axioms.
Restrictions of relations and operators
The restriction concept extends from functions to binary relations. If (R\subseteq A\times A) and (S\subseteq A), the restriction of (R) to (S) is
[ R|_S=R\cap(S\times S). ]
An order relation restricted to a subset remains an order relation of the same general type. In particular, the restriction of a partial order is a partial order, while the restriction of an equivalence relation is an equivalence relation on the selected subset.
For a linear operator (T\colon D(T)\to Y), especially an unbounded operator, restriction may refer to replacement of its domain (D(T)) by a smaller linear subspace. Domain information is essential in operator theory, so two operators with the same formula but different domains can have different spectra, adjoints, or closure properties. An extension of an operator enlarges its domain while preserving its action on the original domain.
Restriction of a representation has a further specialized meaning. If (\pi) is a group representation of (G) and (H) is a subgroup, the restricted representation (\operatorname{Res}^G_H\pi) is obtained by evaluating (\pi) only on elements of (H). This operation underlies the restriction functor, which transfers representations or modules along a homomorphism by retaining the original underlying objects while narrowing the acting algebraic structure.