Exercise 6.1.16

• $$⟨f+h,g⟩=\frac{1}{2\pi }{\int \; <!--\; nolimits\; -->}_0^{2\pi }(f(t)+h(t))\overline{g(t)}\mathit{dt}$$

  $$=\frac{1}{2\pi }{\int \; <!--\; nolimits\; -->}_0^{2\pi }f\overline{g}\mathit{dt}+\frac{1}{2\pi }{\int \; <!--\; nolimits\; -->}_0^{2\pi }(h\overline{g}\mathit{dt}=⟨f,g⟩+⟨h,g⟩.$$

• $$⟨\mathit{cf},g⟩=\frac{1}{2\pi }{\int \; <!--\; nolimits\; -->}_0^{2\pi }\mathit{cf}\overline{g}\mathit{dt}$$

  $$=c(\frac{1}{2\pi }{\int \; <!--\; nolimits\; -->}_0^{2\pi }f\overline{g}\mathit{dt})=c⟨f,g⟩.$$

• $$\overline{⟨f,g⟩}=\overline{\frac{1}{2\pi }{\int \; <!--\; nolimits\; -->}_0^{2\pi }f\overline{g}\mathit{dt}}=\frac{1}{2\pi }{\int \; <!--\; nolimits\; -->}_0^{2\pi }\overline{f\overline{g}}\mathit{dt}$$

  $$=\frac{1}{2\pi }{\int \; <!--\; nolimits\; -->}_0^{2\pi }g\overline{f}\mathit{dt}=⟨g,f⟩.$$

• $$⟨f,f⟩=\frac{1}{2\pi }{\int \; <!--\; nolimits\; -->}_0^{2\pi }\parallel f{\text{∥}}^2\mathit{dt}>0$$

  if $f$ is not zero. Ur... I think this is an exercise for the Adavanced Calculus course.

Then we have that $⟨f,f⟩=0$ but $f\ne 0$.