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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 $f=0$

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    If proving the properties is "very basic" then why are you having to ask somebody else to do it?2012-10-18

1 Answers 1

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$\langle\,f\,,\,g\,\rangle:=\int_a^b f(t)\overline{g(t)}dt=\int_a^b \overline{\overline{f(t)}g(t)}\,dt=\overline{\int_a^b g(t)\overline{f(t)}}\,dt=\overline{\langle\,g\,,\,f\,\rangle}$

$\langle\,f\,,\,f\,\rangle=0\Longleftrightarrow \int_a^b |f(t)|^2\,dt=0\stackrel{\text{by continuity}}\Longleftrightarrow |f(t)|^2=0\,\,\text{on}\,\,[a,b]\Longleftrightarrow f(t)=0$

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    thank you very much DonAntonio..I really appreciate your feed back!!!2012-10-18