Exercise 6.1.13

Check those condition.

• $$⟨x+z,y⟩={⟨x+z,y⟩}_1+{⟨x+z,y⟩}_2$$

  $$={⟨x,y⟩}_1+{⟨z,y⟩}_1+{⟨x,y⟩}_2+{⟨z,y⟩}_2$$

  $$=⟨x,y⟩+⟨z,y⟩.$$

• $$⟨\mathit{cx},y⟩={⟨\mathit{cx},y⟩}_1+{⟨\mathit{cx},y⟩}_2$$

  $$=c{⟨x,y⟩}_1+c{⟨x,y⟩}_2=c⟨x,y⟩.$$

• $$\overline{⟨x,y⟩}=\overline{{⟨x,y⟩}_1}+\overline{{⟨x,y⟩}_2}$$

  $$={⟨y,x⟩}_1+{⟨y,x⟩}_2=⟨y,x⟩.$$

• $$⟨x,x⟩={⟨x,x⟩}_1+{⟨x,x⟩}_2>0$$

  if $x\ne 0$.