Conjugate index

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, two <a href="/facts/Real_number/R02gw5Pb">real numbers</a> 
 
 
 
 p
 ,
 q
 >
 1
 
 
 {\displaystyle p,q>1}
 
 are called conjugate indices (or Hölder conjugates) if

1
            p
          
        
        +
        
          
            1
            q
          
        
        =
        1.
      
    
    {\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1.}

Formally, we also define 
 
 
 
 q
 =
 ∞
 
 
 {\displaystyle q=\infty }
 
 as conjugate to 
 
 
 
 p
 =
 1
 
 
 {\displaystyle p=1}
 
 and <a href="/facts/List_of_Latin_phrases%3a_V/cnbyWbkv">vice versa</a>. 
Conjugate indices are used in <a href="/facts/H%25C3%25B6lder%2527s_inequality/7I1ubTWw">Hölder's inequality</a>, as well as <a href="/facts/Young%2527s_inequality_for_products/d3DnDNsh">Young's inequality for products</a>; the latter can be used to prove the former. If 
 
 
 
 p
 ,
 q
 >
 1
 
 
 {\displaystyle p,q>1}
 
 are conjugate indices, the spaces Lp and Lq are <a href="/facts/Dual_space/4Uw4Knz1">dual</a> to each other (see <a href="/facts/Lp_space/gbad0Rv1">Lp space</a>).

Conjugate index open-in-new

Conjugate index