- #1

RJLiberator

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## Homework Statement

Show that the set {sin(nx)} from n=1 to n=∞ is orthogonal bases for L^2(0, π).

## Homework Equations

## The Attempt at a Solution

Proof: Let f(x)= sin(nx), consider scalar product in L^2(0, π)

[itex]

(ƒ_n , ƒ_m) = \int_{0}^π ƒ_n (x) ƒ_m (x) \, dx = \int_{0}^π sin(nx)sin(mx) \, dx = \frac{1}{2} \int_{0}^π cos(n-m)x - cos(n+m)x \, dx = \frac{1}{2} ( \int_{0}^π cos(n-m)x - \int_{0}^π cos(n+m)x \, dx = \frac{1}{2} ( {\frac{\sin(n-m)x}{(n-m)x}} - {\frac{\sin(n+m)x}{(n+m)x}} ) = 0

[/itex]

Of course the last result is from 0 to pi for both parts, but didn't know how to latex it :p.

My question is, is this everything to show that the set is orthogonal bases for L2 (0, π)?

I think it is, but definitions in the crazy world of PDE are confusing to me. I do not believe I need to show completeness per the question.

Thank you.