Exercise 6.1.28

Check the conditions one by one.

• $$[x+z,y]=\mathit{ℝe}⟨x+z,y⟩=\mathit{ℝe}(⟨x,y⟩+⟨z,y⟩)$$

  $$=\mathit{ℝe}⟨x,y⟩+\mathit{ℝe}⟨z,y⟩=[x,y]+[z,y].$$

• $$[\mathit{cx},y]=\mathit{ℝe}⟨\mathit{cx},y⟩$$

  $$=\mathit{cℝe}⟨x,y⟩=c[x,y],$$

  where $c$ is a real number.

• $$[x,y]=\mathit{ℝe}⟨x,y⟩=\mathit{ℝe}\overline{⟨x,y⟩}$$

  $$=\mathit{ℝe}⟨y,x⟩=[y,x].$$

• $$[x,x]=\mathit{ℝe}⟨x,x⟩=⟨x,x⟩>0$$

  if $x\ne 0$.

Finally, we have $[x,\mathit{ix}]=0$ since

is a pure imaginary number.