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antilinear map

In mathematics, a function f : V → W {\displaystyle f:V\to W} between two complex vector spaces 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 complex number s , {\displaystyle s,} where s ¯ {\displaystyle {\overline {s}}} denotes the complex conjugate of s .

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