Exercise 6.4.20

Check the condition one by one.

• $${⟨x+z,y⟩}^{\prime }=⟨T(x+z),y⟩$$

  $$=⟨T(x),y⟩+⟨T(z),y⟩={⟨x,y⟩}^{\prime }+{⟨z,y⟩}^{\prime }.$$

• $$⟨\mathit{cx},y⟩=⟨T(\mathit{cx}),y⟩$$

  $$=c⟨T(x),y⟩={⟨x,y⟩}^{\prime }.$$

• $$\overline{{⟨x,y⟩}^{\prime }}=\overline{⟨T(x),y⟩}$$

  $$=⟨y,T(x)⟩=⟨T(y),x⟩={⟨y,x⟩}^{\prime }.$$

• $${⟨x,x⟩}^{\prime }=⟨T(x),x⟩>0$$

  if $x$ is not zero.