Prove that $C[a,b]$ is a inner product space..
Please help me to prove the following axioms or can anybody send me a link which includes these proofs.
1.conjugate symmetry property
2.inner product of $f$,$f=0$ iff $f=0$
Prove that $C[a,b]$ is a inner product space..
Please help me to prove the following axioms or can anybody send me a link which includes these proofs.
1.conjugate symmetry property
2.inner product of $f$,$f=0$ iff $f=0$
$\langle\,f\,,\,g\,\rangle:=\int_a^b f(t)\overline{g(t)}dt=\int_a^b \overline{\overline{f(t)}g(t)}\,dt=\overline{\int_a^b g(t)\overline{f(t)}}\,dt=\overline{\langle\,g\,,\,f\,\rangle}$
$\langle\,f\,,\,f\,\rangle=0\Longleftrightarrow \int_a^b |f(t)|^2\,dt=0\stackrel{\text{by continuity}}\Longleftrightarrow |f(t)|^2=0\,\,\text{on}\,\,[a,b]\Longleftrightarrow f(t)=0$