Antilinear map

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
 
 
 
 f
 :
 V
 →
 W
 
 
 {\displaystyle f:V\to W}
 
 between two <a href="/facts/Complex_vector_space/M1pxMLx2">complex vector spaces</a> is said to be antilinear or conjugate-linear if

f
                (
                x
                +
                y
                )
              
              
                
                =
                f
                (
                x
                )
                +
                f
                (
                y
                )
              
              
              
                
                
                   (additivity) 
                
              
            
            
              
                f
                (
                s
                x
                )
              
              
                
                =
                
                  
                    s
                    ¯
                  
                
                f
                (
                x
                )
              
              
              
                
                
                   (conjugate homogeneity) 
                
              
            
          
        
      
    
    {\displaystyle {\begin{alignedat}{9}f(x+y)&=f(x)+f(y)&&\qquad {\text{ (additivity) }}\\f(sx)&={\overline {s}}f(x)&&\qquad {\text{ (conjugate homogeneity) }}\\\end{alignedat}}}

hold for all vectors 
 
 
 
 x
 ,
 y
 ∈
 V
 
 
 {\displaystyle x,y\in V}
 
 and every <a href="/facts/Complex_number/2w7mqI7e">complex number</a> 
 
 
 
 s
 ,
 
 
 {\displaystyle s,}
 
 where 
 
 
 
 
 
 s
 ¯
 
 
 
 
 {\displaystyle {\overline {s}}}
 
 denotes the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> of 
 
 
 
 s
 .
 
 
 {\displaystyle s.}
 
 
Antilinear maps stand in contrast to <a href="/facts/Linear_operator/Q1PBKNVV">linear maps</a>, which are <a href="/facts/Additive_map/lyCB3yy0">additive maps</a> that are <a href="/facts/Homogeneous_map/wF9XX7s5">homogeneous</a> rather than <a href="/facts/Conjugate_homogeneity/wF9XX7s5">conjugate homogeneous</a>. If the vector spaces are <a href="/facts/Real_vector_space/M1pxMLx2">real</a> then antilinearity is the same as linearity.
Antilinear maps occur in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> in the study of <a href="/facts/T-symmetry/Tgf2LJMc">time reversal</a> and in spinor calculus, where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices. Scalar-valued antilinear maps often arise when dealing with <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Inner_product_space/HoyjElEy">inner products</a> and <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a>.

Antilinear map open-in-new

Antilinear map