Exercise 6.1.29

Observe that

So we have an instant property

Now check the conditions one by one.

• $$⟨x+z,y⟩=[x+z,y]+i[x+z,\mathit{iy}]$$

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

• $$⟨(a+\mathit{bi})x,y⟩=[(a+\mathit{bi})x,y]+i[(a+\mathit{bi})x,\mathit{iy}]$$

  $$=[\mathit{ax},y]+[\mathit{bix},y]+i[\mathit{ax},\mathit{iy}]+i[\mathit{bix},\mathit{iy}]$$

  $$=a([x,y]+i[x,\mathit{iy}])+\mathit{bi}([\mathit{ix},\mathit{iy}]-i[\mathit{ix},y])$$

  $$=a⟨x,y⟩+\mathit{bi}([x,y]+i[x,\mathit{iy}])=(a+\mathit{bi})⟨x,y⟩.$$

  Here we use the proven property.

• $$\overline{⟨x,y⟩}=[x,y]-i[x,\mathit{iy}]$$

  $$=[y,x]+i[y,\mathit{ix}]=⟨x,y⟩.$$

  Here we use the proven property again.

• $$⟨x,x⟩=[x,x]+i[x,\mathit{ix}]=[x,x]>0$$

  if $x$ is not zero.