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quasar987

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[tex](f,g)=\int_0^{\pi} fg^*dt[/tex]

an inner product on the space of continuous functions defined on the circle?

We're saying yes because the 3 properties are verified:

i) (af+bg,h) = a(f,h)+b(g,h), as if evident by the properties of the integral.

ii) [tex](f,g)^* = \left( \int_0^{\pi} gf^*dt \right)^* = \int_0^{\pi} (gf^*)^*dt = \int_0^{\pi} fg^*dt = (g,f)[/tex]

iii) [tex](f,f) = \int_0^{\pi} ff^*dt = \int_0^{\pi} |f|^2 dt \geq 0[/tex]

(since |f| >= 0 and = 0 <==> f=0)

Hence all 3 properties are verified. Any objection?