Infinite-dimensional vector function

<h2 id="example">Example</h2>
<p>Set 
  
    
      
        
          f
          
            k
          
        
        (
        t
        )
        =
        t
        
          /
        
        
          k
          
            2
          
        
      
    
    {\displaystyle f_{k}(t)=t/k^{2}}
  
 for every positive <a href="/facts/Integer/PUwwrawx">integer</a> 
  
    
      
        k
      
    
    {\displaystyle k}
  
 and every <a href="/facts/Real_number/R02gw5Pb">real number</a> 
  
    
      
        t
        .
      
    
    {\displaystyle t.}
  
 Then the function 
  
    
      
        f
      
    
    {\displaystyle f}
  
 defined by the formula

f
        (
        t
        )
        =
        (
        
          f
          
            1
          
        
        (
        t
        )
        ,
        
          f
          
            2
          
        
        (
        t
        )
        ,
        
          f
          
            3
          
        
        (
        t
        )
        ,
        …
        )
        
        ,
      
    
    {\displaystyle f(t)=(f_{1}(t),f_{2}(t),f_{3}(t),\ldots )\,,}

takes values that lie in the infinite-dimensional <a href="/facts/Vector_space/M1pxMLx2">vector space</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
 (or 
  
    
      
        
          
            R
          
          
            
              N
            
          
        
      
    
    {\displaystyle \mathbb {R} ^{\mathbb {N} }}
  
) of real-valued <a href="/facts/Sequence/gV2S4uqp">sequences</a>. For example,

f
        (
        2
        )
        =
        
          (
          
            2
            ,
            
              
                2
                4
              
            
            ,
            
              
                2
                9
              
            
            ,
            
              
                2
                16
              
            
            ,
            
              
                2
                25
              
            
            ,
            …
          
          )
        
        .
      
    
    {\displaystyle f(2)=\left(2,{\frac {2}{4}},{\frac {2}{9}},{\frac {2}{16}},{\frac {2}{25}},\ldots \right).}

</p><p>As a number of different <a href="/facts/Topological_space/v2ut7CbK">topologies</a> can be defined on the space 
  
    
      
        X
        ,
      
    
    {\displaystyle X,}
  
 to talk about the <a href="/facts/Derivative/7Ito70Dh">derivative</a> of 
  
    
      
        f
        ,
      
    
    {\displaystyle f,}
  
 it is first necessary to specify a topology on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 or the concept of a <a href="/facts/Limit_(mathematics)/j8JBhj4e">limit</a> in 
  
    
      
        X
        .
      
    
    {\displaystyle X.}

</p><p>Moreover, for any set 
  
    
      
        A
        ,
      
    
    {\displaystyle A,}
  
 there exist infinite-dimensional vector spaces having the (Hamel) <a href="/facts/Dimension_(vector_space)/z8m02A3W">dimension</a> of the <a href="/facts/Cardinality/m6VPojwT">cardinality</a> of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 (for example, the space of functions 
  
    
      
        A
        →
        K
      
    
    {\displaystyle A\to K}
  
 with finitely-many nonzero elements, where 
  
    
      
        K
      
    
    {\displaystyle K}
  
 is the desired <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of scalars). Furthermore, the argument 
  
    
      
        t
      
    
    {\displaystyle t}
  
 could lie in any set instead of the set of real numbers.
</p>
<h2 id="integral-and-derivative">Integral and derivative</h2>
<p>Most theorems on <a href="/facts/Integral/V7lyV997">integration</a> and <a href="/facts/Derivative/7Ito70Dh">differentiation</a> of scalar functions can be generalized to vector-valued functions, often using essentially the same <a href="/facts/Mathematical_proof/7ECDrU80">proofs</a>. Perhaps the most important exception is that <a href="/facts/Absolutely_continuous/NPKDt9Qn">absolutely continuous</a> functions need not equal the integrals of their (a.e.) derivatives (unless, for example, 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a Hilbert space); see <a href="/facts/Radon%25E2%2580%2593Nikodym_theorem/uWepNG2B">Radon–Nikodym theorem</a>
</p><p>A <a href="/facts/Topological_curve/xBQ0pTBW">curve</a> is a <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous</a> map of the <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a> (or more generally, of a non−degenerate <a href="/facts/Closed_interval/e9se3vTe">closed</a> <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> of real numbers) into a <a href="/facts/Topological_space/v2ut7CbK">topological space</a>. An <i><a href="/facts/Arc_(topology)/MkKRCYXT">arc</a></i> is a curve that is also a <a href="/facts/Topological_embedding/HKeVKc42">topological embedding</a>. A curve valued in a <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff space</a> is an arc <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> it is <a href="/facts/Injective_function/5ES6tqf5">injective</a>. 
</p>
<h3>Derivatives</h3>
<p>If 
  
    
      
        f
        :
        [
        0
        ,
        1
        ]
        →
        X
        ,
      
    
    {\displaystyle f:[0,1]\to X,}
  
 where 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a Banach space or another <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> then the <a href="/facts/Derivative/7Ito70Dh">derivative</a> of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 can be defined in the usual way:

f
          ′
        
        (
        t
        )
        =
        
          lim
          
            h
            →
            0
          
        
        
          
            
              f
              (
              t
              +
              h
              )
              −
              f
              (
              t
              )
            
            h
          
        
        .
      
    
    {\displaystyle f'(t)=\lim _{h\to 0}{\frac {f(t+h)-f(t)}{h}}.}

</p>
<h4>Functions with values in a Hilbert space</h4>
<p>If 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is a function of real numbers with values in a Hilbert space 
  
    
      
        X
        ,
      
    
    {\displaystyle X,}
  
 then the derivative of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 at a point 
  
    
      
        t
      
    
    {\displaystyle t}
  
 can be defined as in the finite-dimensional case:

Most results of the finite-dimensional case also hold in the infinite-dimensional case too, with some modifications. Differentiation can also be defined to functions of several variables (for example, 
  
    
      
        t
        ∈
        
          R
          
            n
          
        
      
    
    {\displaystyle t\in R^{n}}
  
 or even 
  
    
      
        t
        ∈
        Y
        ,
      
    
    {\displaystyle t\in Y,}
  
 where 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 is an infinite-dimensional vector space).
</p><p>If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a Hilbert space then any derivative (and any other limit) can be computed componentwise: if

f
        =
        (
        
          f
          
            1
          
        
        ,
        
          f
          
            2
          
        
        ,
        
          f
          
            3
          
        
        ,
        …
        )
      
    
    {\displaystyle f=(f_{1},f_{2},f_{3},\ldots )}

(that is, 
  
    
      
        f
        =
        
          f
          
            1
          
        
        
          e
          
            1
          
        
        +
        
          f
          
            2
          
        
        
          e
          
            2
          
        
        +
        
          f
          
            3
          
        
        
          e
          
            3
          
        
        +
        ⋯
        ,
      
    
    {\displaystyle f=f_{1}e_{1}+f_{2}e_{2}+f_{3}e_{3}+\cdots ,}
  
 where 
  
    
      
        
          e
          
            1
          
        
        ,
        
          e
          
            2
          
        
        ,
        
          e
          
            3
          
        
        ,
        …
      
    
    {\displaystyle e_{1},e_{2},e_{3},\ldots }
  
 is an <a href="/facts/Orthonormal_basis/pzfKHlJG">orthonormal basis</a> of the space 
  
    
      
        X
      
    
    {\displaystyle X}
  
), and 
  
    
      
        
          f
          ′
        
        (
        t
        )
      
    
    {\displaystyle f'(t)}
  
 exists, then

f
          ′
        
        (
        t
        )
        =
        (
        
          f
          
            1
          
          ′
        
        (
        t
        )
        ,
        
          f
          
            2
          
          ′
        
        (
        t
        )
        ,
        
          f
          
            3
          
          ′
        
        (
        t
        )
        ,
        …
        )
        .
      
    
    {\displaystyle f'(t)=(f_{1}'(t),f_{2}'(t),f_{3}'(t),\ldots ).}

However, the existence of a componentwise derivative does not guarantee the existence of a derivative, as componentwise convergence in a Hilbert space does not guarantee convergence with respect to the actual topology of the Hilbert space.
</p><p>Most of the above hold for other <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
 too. However, not as many classical results hold in the <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> setting, for example, an <a href="/facts/Absolutely_continuous/NPKDt9Qn">absolutely continuous</a> function with values in a <a href="/facts/Radon%25E2%2580%2593Nikodym_property/RpzcTu9n">suitable Banach space</a> need not have a derivative anywhere. Moreover, in most Banach spaces setting there are no orthonormal bases.
</p>
<h3>Crinkled arcs</h3>
<p class="note">Main article: <a href="/facts/Crinkled_arc/aYHl5N2k">Crinkled arc</a></p>
<p>If 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
 is an interval contained in the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> of a curve 
  
    
      
        f
      
    
    {\displaystyle f}
  
 that is valued in a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> then the vector 
  
    
      
        f
        (
        b
        )
        −
        f
        (
        a
        )
      
    
    {\displaystyle f(b)-f(a)}
  
 is called the <i>chord of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 determined by 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
</i>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> 
If 
  
    
      
        [
        c
        ,
        d
        ]
      
    
    {\displaystyle [c,d]}
  
 is another interval in its domain then the two chords are said to be <i>non−overlapping chords</i> if 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
 and 
  
    
      
        [
        c
        ,
        d
        ]
      
    
    {\displaystyle [c,d]}
  
 have at most one end−point in common.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> 
Intuitively, two non−overlapping chords of a curve valued in an <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a> are <a href="/facts/Orthogonal_vectors/HoyjElEy">orthogonal vectors</a> if the curve makes a <a href="/facts/Right_angle/psaslRq7">right angle</a> turn somewhere along its path between its starting point and its ending point. 
If every pair of non−overlapping chords are orthogonal then such a right turn happens at every point of the curve; such a curve can not be <a href="/facts/Differentiable/jQqTgLk1">differentiable</a> at any point.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> 
A <i>crinkled arc</i> is an injective continuous curve with the property that any two non−overlapping chords are orthogonal vectors. 
An example of a crinkled arc in the <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert</a> <a href="/facts/Lp_space/gbad0Rv1">
  
    
      
        
          L
          
            2
          
        
      
    
    {\displaystyle L^{2}}
  
 space</a> 
  
    
      
        
          L
          
            2
          
        
        (
        0
        ,
        1
        )
      
    
    {\displaystyle L^{2}(0,1)}
  
 is:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>

f
                :
                
              
              
              
                [
                0
                ,
                1
                ]
              
              
              
                
                →
                
              
              
                
                  L
                  
                    2
                  
                
                (
                0
                ,
                1
                )
              
            
            
              
              
              
                t
              
              
              
                
                ↦
                
              
              
                
                  
                    1
                  
                  
                    [
                    0
                    ,
                    t
                    ]
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{alignedat}{4}f:\;&&[0,1]&&\;\to \;&L^{2}(0,1)\\[0.3ex]&&t&&\;\mapsto \;&\mathbb {1} _{[0,t]}\\\end{alignedat}}}

where 
  
    
      
        
          
            1
          
          
            [
            0
            ,
            
            t
            ]
          
        
        :
        (
        0
        ,
        1
        )
        →
        {
        0
        ,
        1
        }
      
    
    {\displaystyle \mathbb {1} _{[0,\,t]}:(0,1)\to \{0,1\}}
  
 is the <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> defined by

x
        
        ↦
        
        
          
            {
            
              
                
                  1
                
                
                  
                     if 
                  
                  x
                  ∈
                  [
                  0
                  ,
                  t
                  ]
                
              
              
                
                  0
                
                
                  
                     otherwise 
                  
                
              
            
            
          
        
      
    
    {\displaystyle x\;\mapsto \;{\begin{cases}1&{\text{ if }}x\in [0,t]\\0&{\text{ otherwise }}\end{cases}}}

A crinkled arc can be found in every infinite−dimensional Hilbert space because any such space contains a <a href="/facts/Closed_set/upYnRTUj">closed</a> <a href="/facts/Vector_subspace/2wtKTgHL">vector subspace</a> that is <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to 
  
    
      
        
          L
          
            2
          
        
        (
        0
        ,
        1
        )
        .
      
    
    {\displaystyle L^{2}(0,1).}
  
<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> 
A crinkled arc 
  
    
      
        f
        :
        [
        0
        ,
        1
        ]
        →
        X
      
    
    {\displaystyle f:[0,1]\to X}
  
 is said to be <i>normalized</i> if 
  
    
      
        f
        (
        0
        )
        =
        0
        ,
      
    
    {\displaystyle f(0)=0,}
  
 
  
    
      
        ‖
        f
        (
        1
        )
        ‖
        =
        1
        ,
      
    
    {\displaystyle \|f(1)\|=1,}
  
 and the <a href="/facts/Span_(linear_algebra)/vPjclEtb">span</a> of its <a href="/facts/Image_(mathematics)/KANefMAl">image</a> 
  
    
      
        f
        (
        [
        0
        ,
        1
        ]
        )
      
    
    {\displaystyle f([0,1])}
  
 is a <a href="/facts/Dense_set/nPT9Yxao">dense subset</a> of 
  
    
      
        X
        .
      
    
    {\displaystyle X.}
  
<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>

<p>Proposition<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>—Given any two normalized crinkled arcs in a Hilbert space, each is unitarily equivalent to a reparameterization of the other.
</p>

<p>If 
  
    
      
        h
        :
        [
        0
        ,
        1
        ]
        →
        [
        0
        ,
        1
        ]
      
    
    {\displaystyle h:[0,1]\to [0,1]}
  
 is an <a href="/facts/Increasing_function/MjQK0YL2">increasing</a> <a href="/facts/Homeomorphism/jMfWg3Mn">homeomorphism</a> then 
  
    
      
        f
        ∘
        h
      
    
    {\displaystyle f\circ h}
  
 is called a <i>reparameterization</i> of the curve 
  
    
      
        f
        :
        [
        0
        ,
        1
        ]
        →
        X
        .
      
    
    {\displaystyle f:[0,1]\to X.}
  
<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> 
Two curves 
  
    
      
        f
      
    
    {\displaystyle f}
  
 and 
  
    
      
        g
      
    
    {\displaystyle g}
  
 in an <a href="/facts/Inner_product_space/HoyjElEy">inner product space</a> 
  
    
      
        X
      
    
    {\displaystyle X}
  
 are <i>unitarily equivalent</i> if there exists a <a href="/facts/Unitary_operator/rkypfQQ9">unitary operator</a> 
  
    
      
        L
        :
        X
        →
        X
      
    
    {\displaystyle L:X\to X}
  
 (which is an <a href="/facts/Isometry/K5wX8ah6">isometric</a> <a href="/facts/Linear_map/Q1PBKNVV">linear</a> <a href="/facts/Bijection/j4gTuTmW">bijection</a>) such that 
  
    
      
        g
        =
        L
        ∘
        f
      
    
    {\displaystyle g=L\circ f}
  
 (or equivalently, 
  
    
      
        f
        =
        
          L
          
            −
            1
          
        
        ∘
        g
      
    
    {\displaystyle f=L^{-1}\circ g}
  
).
</p>
<h3>Measurability</h3>
<p>The <a href="/facts/Measurable_function/NgIHnb1b">measurability</a> of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 can be defined by a number of ways, most important of which are <a href="/facts/Bochner_measurability/HxXFeAgt">Bochner measurability</a> and <a href="/facts/Weak_measurability/cAFxsUvL">weak measurability</a>.
</p>
<h3>Integrals</h3>
<p>The most important integrals of 
  
    
      
        f
      
    
    {\displaystyle f}
  
 are called <a href="/facts/Bochner_integral/RpzcTu9n">Bochner integral</a> (when 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a Banach space) and <a href="/facts/Pettis_integral/NLPpIDsf">Pettis integral</a> (when 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is a topological vector space). Both these integrals commute with <a href="/facts/Linear_functional/oGh7Asrz">linear functionals</a>. Also 
  
    
      
        
          L
          
            p
          
        
      
    
    {\displaystyle L^{p}}
  
 spaces have been <a href="/facts/Bochner_space/tSuPabsJ">defined</a> for such functions.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Differentiation_in_Fr%25C3%25A9chet_spaces/fXqMCIx8">Differentiation in Fréchet spaces</a></li>
<li><a href="/facts/Differentiable_vector%25E2%2580%2593valued_functions_from_Euclidean_space/FyAaenMe">Differentiable vector–valued functions from Euclidean space</a> – Differentiable function in functional analysisPages displaying short descriptions of redirect targets</li></ul>

<ul><li>Einar Hille & Ralph Phillips: "Functional Analysis and Semi Groups", Amer. Math. Soc. Colloq. Publ. Vol. 31, Providence, R.I., 1957.</li>
<li><a href="/facts/Paul_Richard_Halmos/mR11lcap">Halmos, Paul R.</a> (8 November 1982). <i>A Hilbert Space Problem Book</i>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 19 (2nd ed.). New York: <a href="/facts/Springer_Publishing/JfRev59b">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-90685-0. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/8169781">8169781</a>.</li></ul>

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

<ol>
<li id="fn:1"><p>Halmos 1982, pp. 5−7. - Halmos, Paul R. (8 November 1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90685-0. OCLC 8169781. <a href="https://search.worldcat.org/oclc/8169781" target="_blank">https://search.worldcat.org/oclc/8169781</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Halmos 1982, pp. 5−7. - Halmos, Paul R. (8 November 1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90685-0. OCLC 8169781. <a href="https://search.worldcat.org/oclc/8169781" target="_blank">https://search.worldcat.org/oclc/8169781</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Halmos 1982, pp. 5−7. - Halmos, Paul R. (8 November 1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90685-0. OCLC 8169781. <a href="https://search.worldcat.org/oclc/8169781" target="_blank">https://search.worldcat.org/oclc/8169781</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Halmos 1982, pp. 5−7, 168−170. - Halmos, Paul R. (8 November 1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90685-0. OCLC 8169781. <a href="https://search.worldcat.org/oclc/8169781" target="_blank">https://search.worldcat.org/oclc/8169781</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Halmos 1982, pp. 5−7, 168−170. - Halmos, Paul R. (8 November 1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90685-0. OCLC 8169781. <a href="https://search.worldcat.org/oclc/8169781" target="_blank">https://search.worldcat.org/oclc/8169781</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Halmos 1982, pp. 5−7, 168−170. - Halmos, Paul R. (8 November 1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90685-0. OCLC 8169781. <a href="https://search.worldcat.org/oclc/8169781" target="_blank">https://search.worldcat.org/oclc/8169781</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Halmos 1982, pp. 5−7, 168−170. - Halmos, Paul R. (8 November 1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90685-0. OCLC 8169781. <a href="https://search.worldcat.org/oclc/8169781" target="_blank">https://search.worldcat.org/oclc/8169781</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Halmos 1982, pp. 5−7. - Halmos, Paul R. (8 November 1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90685-0. OCLC 8169781. <a href="https://search.worldcat.org/oclc/8169781" target="_blank">https://search.worldcat.org/oclc/8169781</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

Infinite-dimensional vector function open-in-new

Infinite-dimensional vector function