证: Vα,βeV,vfev*,VkePp(α+β)(f) = (α+β)**(f)= f(α+β)= f(α)+ f(β)=α**(f)+ β*(f) = (α)(f)+Φ(β)(f):. p(α+ β)(f) =β(α)+(β)同理p(kα) = kp(α)p(kα)(f)= (kα)**(f)= f(kα) = kf(α)= kα*(f) = kp(α)(f)所以?保持加法和数量乘法810.2对偶空间区区§10.2 对偶空间 证： *        , , , V f V k P **      ( )( ) ( ) ( ) + = + f f = + = + f f f ( ) ( ) ( )     ** ** = + = +       ( ) ( ) ( )( ) ( )( ) f f f f  + = +        ( )( ) ( ) ( ) f 同理     ( ) ( ) k k = **      ( )( ) ( ) ( ) ( ) ( ) k f k f f k kf = = = ** = = k f k f    ( ) ( )( ) 所以 保持加法和数量乘法