Differentiable vector–valued functions from Euclidean space

<p>In the mathematical discipline of <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, a differentiable vector-valued function from Euclidean space is a <a href="/facts/Differentiable/jQqTgLk1">differentiable</a> function valued in a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> (TVS) whose <a href="/facts/Domain_of_a_function/FQvoeUpX">domains</a> is a subset of some <a href="/facts/Dimension_(vector_space)/z8m02A3W">finite-dimensional</a> <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>. 
It is possible to generalize the notion of <a href="/facts/Derivative_(mathematics)/7Ito70Dh">derivative</a> to functions whose domain and codomain are subsets of arbitrary <a href="/facts/Topological_vector_space/XRex1ChN">topological vector spaces</a> (TVSs) in multiple ways. 
But when the domain of a TVS-valued function is a subset of a finite-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> then many of these notions become <a href="/facts/Logically_equivalent/ysCWmsxA">logically equivalent</a> resulting in a much more limited number of generalizations of the derivative and additionally, differentiability is also more <a href="/facts/Well-behaved/KmKHjJ1H">well-behaved</a> compared to the general case. 
This article presents the theory of 
  
    
      
        k
      
    
    {\displaystyle k}
  
-times continuously differentiable functions on an open subset 
  
    
      
        Ω
      
    
    {\displaystyle \Omega }
  
 of Euclidean space 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
 (
  
    
      
        1
        ≤
        n
        <
        ∞
      
    
    {\displaystyle 1\leq n<\infty }
  
), which is an important special case of <a href="/facts/Differentiation_(mathematics)/7Ito70Dh">differentiation</a> between arbitrary TVSs. 
This importance stems partially from the fact that every finite-dimensional vector subspace of a Hausdorff topological vector space is <a href="/facts/TVS_isomorphism/XRex1ChN">TVS isomorphic</a> to Euclidean space 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
 so that, for example, this special case can be applied to any function whose domain is an arbitrary Hausdorff TVS by <a href="/facts/Restriction_of_a_function/JV59DhKy">restricting it</a> to finite-dimensional vector subspaces. 
</p><p>All vector spaces will be assumed to be over the field 
  
    
      
        
          F
        
        ,
      
    
    {\displaystyle \mathbb {F} ,}
  
 where 
  
    
      
        
          F
        
      
    
    {\displaystyle \mathbb {F} }
  
 is either the <a href="/facts/Real_numbers/R02gw5Pb">real numbers</a> 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 or the <a href="/facts/Complex_numbers/2w7mqI7e">complex numbers</a> 
  
    
      
        
          C
        
        .
      
    
    {\displaystyle \mathbb {C} .}

</p>

Differentiable vector–valued functions from Euclidean space open-in-new

Differentiable vector–valued functions from Euclidean space