Continuous embedding

<h2 id="definition">Definition</h2>
Let X and Y be two normed vector spaces, with norms ||·||X and ||·||Y respectively, such that X ⊆ Y. If the <a href="/facts/Identity_function/9AtsP0LC">inclusion map (identity function)</a>

i
        :
        X
        ↪
        Y
        :
        x
        ↦
        x
      
    
    {\displaystyle i:X\hookrightarrow Y:x\mapsto x}

is continuous, i.e. if there exists a constant C > 0 such that

‖
        x
        
          ‖
          
            Y
          
        
        ≤
        C
        ‖
        x
        
          ‖
          
            X
          
        
      
    
    {\displaystyle \|x\|_{Y}\leq C\|x\|_{X}}

for every x in X, then X is said to be continuously embedded in Y. Some authors use the hooked arrow "↪" to denote a continuous embedding, i.e. "X ↪ Y" means "X and Y are normed spaces with X continuously embedded in Y". This is a consistent use of notation from the point of view of the <a href="/facts/Category_of_topological_vector_spaces/BvHfNnrx">category of topological vector spaces</a>, in which the <a href="/facts/Morphism/Kdpuyz3S">morphisms</a> ("arrows") are the <a href="/facts/Continuous_linear_map/KcqjDBrJ">continuous linear maps</a>.

<h2 id="examples">Examples</h2>
<ul><li>A finite-dimensional example of a continuous embedding is given by a natural embedding of the <a href="/facts/Real_line/6eq42xX5">real line</a> X = R into the plane Y = R2, where both spaces are given the Euclidean norm:</li></ul>

i
        :
        
          R
        
        →
        
          
            R
          
          
            2
          
        
        :
        x
        ↦
        (
        x
        ,
        0
        )
      
    
    {\displaystyle i:\mathbf {R} \to \mathbf {R} ^{2}:x\mapsto (x,0)}

In this case, ||x||X = ||x||Y for every real number X. Clearly, the optimal choice of constant C is C = 1.
<ul><li>An infinite-dimensional example of a continuous embedding is given by the <a href="/facts/Rellich%25E2%2580%2593Kondrachov_theorem/DL2Ct0Wj">Rellich–Kondrachov theorem</a>: let Ω ⊆ Rn be an <a href="/facts/Open_set/QfTCIqMu">open</a>, <a href="/facts/Bounded_set/yCldjtoy">bounded</a>, <a href="/facts/Lipschitz_domain/kWcNp9mK">Lipschitz domain</a>, and let 1 ≤ p < n. Set</li></ul>

p
          
            ∗
          
        
        =
        
          
            
              n
              p
            
            
              n
              −
              p
            
          
        
        .
      
    
    {\displaystyle p^{*}={\frac {np}{n-p}}.}

Then the Sobolev space W1,p(Ω; R) is continuously embedded in the <a href="/facts/Lp_space/gbad0Rv1">Lp space</a> Lp∗(Ω; R). In fact, for 1 ≤ q < p∗, this embedding is <a href="/facts/Compactly_embedded/6mMeyhdG">compact</a>. The optimal constant C will depend upon the geometry of the domain Ω.
<ul><li>Infinite-dimensional spaces also offer examples of discontinuous embeddings. For example, consider</li></ul>

X
        =
        Y
        =
        
          C
          
            0
          
        
        (
        [
        0
        ,
        1
        ]
        ;
        
          R
        
        )
        ,
      
    
    {\displaystyle X=Y=C^{0}([0,1];\mathbf {R} ),}

the space of continuous real-valued functions defined on the unit interval, but equip X with the L1 norm and Y with the <a href="/facts/Supremum_norm/5yIkjNE3">supremum norm</a>. For n ∈ N, let fn be the <a href="/facts/Continuous_function/sKbl02pB">continuous</a>, <a href="/facts/Piecewise_linear_function/6bKlCbEa">piecewise linear function</a> given by

f
          
            n
          
        
        (
        x
        )
        =
        
          
            {
            
              
                
                  −
                  
                    n
                    
                      2
                    
                  
                  x
                  +
                  n
                  ,
                
                
                  0
                  ≤
                  x
                  ≤
                  
                    
                      
                        1
                        n
                      
                    
                  
                  ;
                
              
              
                
                  0
                  ,
                
                
                  
                    otherwise.
                  
                
              
            
            
          
        
      
    
    {\displaystyle f_{n}(x)={\begin{cases}-n^{2}x+n,&0\leq x\leq {\tfrac {1}{n}};\\0,&{\text{otherwise.}}\end{cases}}}

Then, for every n, ||fn||Y = ||fn||∞ = n, but

‖
        
          f
          
            n
          
        
        
          ‖
          
            
              L
              
                1
              
            
          
        
        =
        
          ∫
          
            0
          
          
            1
          
        
        
          |
        
        
          f
          
            n
          
        
        (
        x
        )
        
          |
        
        
        
          d
        
        x
        =
        
          
            1
            2
          
        
        .
      
    
    {\displaystyle \|f_{n}\|_{L^{1}}=\int _{0}^{1}|f_{n}(x)|\,\mathrm {d} x={\frac {1}{2}}.}

Hence, no constant C can be found such that ||fn||Y ≤ C||fn||X, and so the embedding of X into Y is discontinuous.
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Compact_embedding/6mMeyhdG">Compact embedding</a></li></ul>

<ul><li>Renardy, M. & Rogers, R.C. (1992). An Introduction to Partial Differential Equations. Springer-Verlag, Berlin. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-97952-2.</li></ul>

Continuous embedding open-in-new

Continuous embedding