Function space

<h2 id="in-linear-algebra">In linear algebra</h2>
See also: <a href="/facts/Vector_space/M1pxMLx2">Vector space § Function spaces</a>

Let F be a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> and let X be any set. The functions X → F can be given the structure of a vector space over F where the operations are defined pointwise, that is, for any f, g : X → F, any x in X, and any c in F, define

(
                f
                +
                g
                )
                (
                x
                )
              
              
                
                =
                f
                (
                x
                )
                +
                g
                (
                x
                )
              
            
            
              
                (
                c
                ⋅
                f
                )
                (
                x
                )
              
              
                
                =
                c
                ⋅
                f
                (
                x
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(c\cdot f)(x)&=c\cdot f(x)\end{aligned}}}

When the domain X has additional structure, one might consider instead the <a href="/facts/Subset/SJGERmbA">subset</a> (or <a href="/facts/Linear_subspace/2wtKTgHL">subspace</a>) of all such functions which respect that structure. For example, if V and also X itself are vector spaces over F, the set of <a href="/facts/Linear_map/Q1PBKNVV">linear maps</a> X → V form a vector space over F with pointwise operations (often denoted <a href="/facts/Hom_set/Kdpuyz3S">Hom</a>(X,V)). One such space is the <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> of X: the set of <a href="/facts/Linear_form/oGh7Asrz">linear functionals</a> X → F with addition and scalar multiplication defined pointwise.
The cardinal <a href="/facts/Dimension/0ktmUyac">dimension</a> of a function space with no extra structure can be found by the <a href="/facts/Erd%25C5%2591s%25E2%2580%2593Kaplansky_theorem/JsVd6zra">Erdős–Kaplansky theorem</a>.

<h2 id="examples">Examples</h2>
Function spaces appear in various areas of mathematics:

<ul><li>In <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, the set of functions from X to Y may be denoted {X → Y} or YX.
<ul><li>As a special case, the <a href="/facts/Power_set/FVuf8PDD">power set</a> of a set X may be identified with the set of all functions from X to {0, 1}, denoted 2X.</li></ul></li>
<li>The set of <a href="/facts/Bijection/j4gTuTmW">bijections</a> from X to Y is denoted 
 
 
 
 X
 ↔
 Y
 
 
 {\displaystyle X\leftrightarrow Y}
 
. The factorial notation X! may be used for permutations of a single set X.</li>
<li>In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, the same is seen for <a href="/facts/Continuous_function/sKbl02pB">continuous</a> linear transformations, including <a href="/facts/Topological_vector_space/XRex1ChN">topologies on the vector spaces</a> in the above, and many of the major examples are function spaces carrying a <a href="/facts/Topology/2EqW47cU">topology</a>; the best known examples include <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a> and <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a>.</li>
<li>In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, the set of all functions from the <a href="/facts/Natural_number/u65og8JE">natural numbers</a> to some set X is called a <a href="/facts/Sequence_space/2HkcM12K">sequence space</a>. It consists of the set of all possible <a href="/facts/Sequences/gV2S4uqp">sequences</a> of elements of X.</li>
<li>In <a href="/facts/Topology/2EqW47cU">topology</a>, one may attempt to put a topology on the space of continuous functions from a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> X to another one Y, with utility depending on the nature of the spaces. A commonly used example is the <a href="/facts/Compact-open_topology/qMGsW8gk">compact-open topology</a>, e.g. <a href="/facts/Loop_space/S5xkdCkx">loop space</a>. Also available is the <a href="/facts/Product_topology/0hg02IaX">product topology</a> on the space of set theoretic functions (i.e. not necessarily continuous functions) YX. In this context, this topology is also referred to as the <a href="/facts/Topology_of_pointwise_convergence/TXGpeyIK">topology of pointwise convergence</a>.</li>
<li>In <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, the study of <a href="/facts/Homotopy_theory/MJwYyRny">homotopy theory</a> is essentially that of discrete invariants of function spaces;</li>
<li>In the theory of <a href="/facts/Stochastic_process/Ng1n1dnB">stochastic processes</a>, the basic technical problem is how to construct a <a href="/facts/Probability_measure/8BjhgoPR">probability measure</a> on a function space of paths of the process (functions of time);</li>
<li>In <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, the function space is called an <a href="/facts/Exponential_object/4MYWRF9h">exponential object</a> or <a href="/facts/Exponential_object/4MYWRF9h">map object</a>. It appears in one way as the representation <a href="/facts/Canonical_bifunctor/H76BLKCD">canonical bifunctor</a>; but as (single) functor, of type 
 
 
 
 [
 X
 ,
 −
 ]
 
 
 {\displaystyle [X,-]}
 
, it appears as an <a href="/facts/Adjoint_functor/bJ1P7AN2">adjoint functor</a> to a functor of type 
 
 
 
 −
 ×
 X
 
 
 {\displaystyle -\times X}
 
 on objects;</li>
<li>In <a href="/facts/Functional_programming/cdxqPEIM">functional programming</a> and <a href="/facts/Lambda_calculus/DaD3RuQ4">lambda calculus</a>, <a href="/facts/Function_type/CmbusBzk">function types</a> are used to express the idea of <a href="/facts/Higher-order_function/VjnjGbpt">higher-order functions</a></li>
<li>In programming more generally, many <a href="/facts/Higher-order_function/VjnjGbpt">higher-order function</a> concepts occur with or without explicit typing, such as <a href="/facts/Closure_(computer_programming)/droP22Xn">closures</a>.</li>
<li>In <a href="/facts/Domain_theory/G1wRMK2K">domain theory</a>, the basic idea is to find constructions from <a href="/facts/Partial_order/qb4a3FAX">partial orders</a> that can model lambda calculus, by creating a well-behaved <a href="/facts/Cartesian_closed_category/mPXAQps2">Cartesian closed category</a>.</li>
<li>In the <a href="/facts/Representation_theory_of_finite_groups/WNnRPRAK">representation theory of finite groups</a>, given two finite-dimensional representations V and W of a group G, one can form a representation of G over the vector space of linear maps Hom(V,W) called the <a href="/facts/Hom_representation/RXEV8te8">Hom representation</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a></li></ul>
<h2 id="functional-analysis">Functional analysis</h2>
<a href="/facts/Functional_analysis/MIp233MT">Functional analysis</a> is organized around adequate techniques to bring function spaces as <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a> within reach of the ideas that would apply to <a href="/facts/Normed_space/rmDljfyE">normed spaces</a> of finite dimension. Here we use the real line as an example domain, but the spaces below exist on suitable open subsets 
 
 
 
 Ω
 ⊆
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \Omega \subseteq \mathbb {R} ^{n}}

<ul><li>
 
 
 
 C
 (
 
 R
 
 )
 
 
 {\displaystyle C(\mathbb {R} )}
 
 <a href="/facts/Continuous_functions/sKbl02pB">continuous functions</a> endowed with the <a href="/facts/Uniform_norm/5yIkjNE3">uniform norm</a> topology</li>
<li>
 
 
 
 
 C
 
 c
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle C_{c}(\mathbb {R} )}
 
 continuous functions with <a href="/facts/Support_(mathematics)/BTuMGbsb">compact support</a></li>
<li>
 
 
 
 B
 (
 
 R
 
 )
 
 
 {\displaystyle B(\mathbb {R} )}
 
 <a href="/facts/Bounded_function/whVjbtuk">bounded functions</a></li>
<li>
 
 
 
 
 C
 
 0
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle C_{0}(\mathbb {R} )}
 
 continuous functions which vanish at infinity</li>
<li>
 
 
 
 
 C
 
 r
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle C^{r}(\mathbb {R} )}
 
 continuous functions that have r continuous derivatives.</li>
<li>
 
 
 
 
 C
 
 ∞
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle C^{\infty }(\mathbb {R} )}
 
 <a href="/facts/Smooth_functions/mffL4ch2">smooth functions</a></li>
<li>
 
 
 
 
 C
 
 c
 
 
 ∞
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle C_{c}^{\infty }(\mathbb {R} )}
 
 <a href="/facts/Smooth_functions/mffL4ch2">smooth functions</a> with <a href="/facts/Support_(mathematics)/BTuMGbsb">compact support</a> (i.e. the set of <a href="/facts/Bump_function/ExdBtMHN">bump functions</a>)</li>
<li>
 
 
 
 
 C
 
 ω
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle C^{\omega }(\mathbb {R} )}
 
 <a href="/facts/Analytic_function/JhL3z8FP">real analytic functions</a></li>
<li>
 
 
 
 
 L
 
 p
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle L^{p}(\mathbb {R} )}
 
, for 
 
 
 
 1
 ≤
 p
 ≤
 ∞
 
 
 {\displaystyle 1\leq p\leq \infty }
 
, is the <a href="/facts/Lp_space/gbad0Rv1">Lp space</a> of <a href="/facts/Measurable_function/NgIHnb1b">measurable</a> functions whose p-norm 
 
 
 
 ‖
 f
 
 ‖
 
 p
 
 
 =
 
 
 (
 
 
 ∫
 
 
 R
 
 
 
 
 |
 
 f
 
 
 |
 
 
 p
 
 
 
 )
 
 
 1
 
 /
 
 p
 
 
 
 
 {\textstyle \|f\|_{p}=\left(\int _{\mathbb {R} }|f|^{p}\right)^{1/p}}
 
 is finite</li>
<li>
 
 
 
 
 
 S
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle {\mathcal {S}}(\mathbb {R} )}
 
, the <a href="/facts/Schwartz_space/B035faw7">Schwartz space</a> of <a href="/facts/Rapidly_decreasing/nCAl6pYS">rapidly decreasing</a> <a href="/facts/Smooth_functions/mffL4ch2">smooth functions</a> and its continuous dual, 
 
 
 
 
 
 
 S
 
 
 ′
 
 (
 
 R
 
 )
 
 
 {\displaystyle {\mathcal {S}}'(\mathbb {R} )}
 
 <a href="/facts/Tempered_distributions/yfaJ4mUp">tempered distributions</a></li>
<li>
 
 
 
 D
 (
 
 R
 
 )
 
 
 {\displaystyle D(\mathbb {R} )}
 
 compact support in limit topology</li>
<li>
 
 
 
 
 W
 
 k
 ,
 p
 
 
 
 
 {\displaystyle W^{k,p}}
 
 <a href="/facts/Sobolev_space/34qxWCyr">Sobolev space</a> of functions whose <a href="/facts/Weak_derivative/oehSmSMg">weak derivatives</a> up to order k are in 
 
 
 
 
 L
 
 p
 
 
 
 
 {\displaystyle L^{p}}
 
</li>
<li>
 
 
 
 
 
 
 O
 
 
 
 U
 
 
 
 
 {\displaystyle {\mathcal {O}}_{U}}
 
 holomorphic functions</li>
<li>linear functions</li>
<li>piecewise linear functions</li>
<li>continuous functions, compact open topology</li>
<li>all functions, space of pointwise convergence</li>
<li><a href="/facts/Hardy_space/aN45dgx7">Hardy space</a></li>
<li><a href="/facts/H%25C3%25B6lder_space/CI0yp6kD">Hölder space</a></li>
<li><a href="/facts/C%25C3%25A0dl%25C3%25A0g/R9Jz9QLh">Càdlàg</a> functions, also known as the <a href="/facts/Anatoliy_Skorokhod/boIrt8wI">Skorokhod</a> space</li>
<li>
 
 
 
 
 
 Lip
 
 
 0
 
 
 (
 
 R
 
 )
 
 
 {\displaystyle {\text{Lip}}_{0}(\mathbb {R} )}
 
, the space of all <a href="/facts/Lipschitz_continuous/Hw20EPEU">Lipschitz</a> functions on 
 
 
 
 
 R
 
 
 
 {\displaystyle \mathbb {R} }
 
 that vanish at zero.</li></ul>
<h2 id="norm">Norm</h2>
If y is an element of the function space 
 
 
 
 
 
 C
 
 
 (
 a
 ,
 b
 )
 
 
 {\displaystyle {\mathcal {C}}(a,b)}
 
 of all <a href="/facts/Continuous_function/sKbl02pB">continuous functions</a> that are defined on a <a href="/facts/Closed_interval/e9se3vTe">closed interval</a> [a, b], the <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> 
 
 
 
 ‖
 y
 
 ‖
 
 ∞
 
 
 
 
 {\displaystyle \|y\|_{\infty }}
 
 defined on 
 
 
 
 
 
 C
 
 
 (
 a
 ,
 b
 )
 
 
 {\displaystyle {\mathcal {C}}(a,b)}
 
 is the maximum <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> of y (x) for a ≤ x ≤ b,<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

‖
        y
        
          ‖
          
            ∞
          
        
        ≡
        
          max
          
            a
            ≤
            x
            ≤
            b
          
        
        
          |
        
        y
        (
        x
        )
        
          |
        
        
        
          where
        
         
         
        y
        ∈
        
          
            C
          
        
        (
        a
        ,
        b
        )
      
    
    {\displaystyle \|y\|_{\infty }\equiv \max _{a\leq x\leq b}|y(x)|\qquad {\text{where}}\ \ y\in {\mathcal {C}}(a,b)}

is called the <a href="/facts/Uniform_norm/5yIkjNE3">uniform norm</a> or supremum norm ('sup norm').

<h2 id="bibliography">Bibliography</h2>
<ul><li>Kolmogorov, A. N., & Fomin, S. V. (1967). Elements of the theory of functions and functional analysis. Courier Dover Publications.</li>
<li>Stein, Elias; Shakarchi, R. (2011). Functional Analysis: An Introduction to Further Topics in Analysis. Princeton University Press.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/List_of_mathematical_functions/bua9ERtP">List of mathematical functions</a></li>
<li><a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebra</a></li>
<li><a href="/facts/Tensor_field/cAC6F6rF">Tensor field</a></li>
<li><a href="/facts/Spectral_theory/8sAEEIMM">Spectral theory</a></li>
<li><a href="/facts/Functional_determinant/preMcoEw">Functional determinant</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Fulton, William; Harris, Joe (1991). Representation Theory: A First Course. Springer Science & Business Media. p. 4. ISBN 9780387974958. <a href="9780387974958" target="_blank">9780387974958</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Gelfand, I. M.; Fomin, S. V. (2000). Silverman, Richard A. (ed.). Calculus of variations (Unabridged repr. ed.). Mineola, New York: Dover Publications. p. 6. ISBN 978-0486414485. <a href="978-0486414485" target="_blank">978-0486414485</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Function space open-in-new

Function space