Uniformly smooth space

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a uniformly smooth space is a <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 satisfying the property that for every 
 
 
 
 ϵ
 >
 0
 
 
 {\displaystyle \epsilon >0}
 
 there exists 
 
 
 
 δ
 >
 0
 
 
 {\displaystyle \delta >0}
 
 such that if 
 
 
 
 x
 ,
 y
 ∈
 X
 
 
 {\displaystyle x,y\in X}
 
 with 
 
 
 
 ‖
 x
 ‖
 =
 1
 
 
 {\displaystyle \|x\|=1}
 
 and 
 
 
 
 ‖
 y
 ‖
 ≤
 δ
 
 
 {\displaystyle \|y\|\leq \delta }
 
 then

‖
        x
        +
        y
        ‖
        +
        ‖
        x
        −
        y
        ‖
        ≤
        2
        +
        ϵ
        ‖
        y
        ‖
        .
      
    
    {\displaystyle \|x+y\|+\|x-y\|\leq 2+\epsilon \|y\|.}

The modulus of smoothness of a normed space X is the function ρX defined for every t > 0 by the formula

ρ
          
            X
          
        
        (
        t
        )
        =
        sup
        
          
            {
          
        
        
          
            
              ‖
              x
              +
              y
              ‖
              +
              ‖
              x
              −
              y
              ‖
            
            2
          
        
        −
        1
        
        :
        
        ‖
        x
        ‖
        =
        1
        ,
        
        ‖
        y
        ‖
        =
        t
        
          
            }
          
        
        .
      
    
    {\displaystyle \rho _{X}(t)=\sup {\Bigl \{}{\frac {\|x+y\|+\|x-y\|}{2}}-1\,:\,\|x\|=1,\;\|y\|=t{\Bigr \}}.}

The triangle inequality yields that ρX(t ) ≤ t. The normed space X is uniformly smooth if and only if ρX(t ) / t tends to 0 as t tends to 0.

Uniformly smooth space open-in-new

Uniformly smooth space