Let $V = \text{span}(1, \cos x,..., \cos nx, \sin x,...,\sin nx)$ for a fixed positive $n$.
In the inner product space of real-valued, continuous functions on $[-\pi,\pi]$ with inner-product
$$\langle f,g\rangle= \int_{-\pi}^{\pi}f(x)g(x)dx$$
Define $D\in \mathcal L(V)$ by $Df=f'$
Show $D^*=-D$
In this case:
$$\langle Tf(x),g(x)\rangle=\langle f(x),T^*g(x)\rangle$$ and specifically $$\langle Df(x),g(x)\rangle=\langle f(x),-D g(x)\rangle$$
What I have done so far: I took $f(x)= a_0+a_1\sin x+a_2\cos x$ and $g(x)=b_0+b_1\sin x+b_2\cos x$ and applied the above-mentioned inner-products to see an example of how things worked.
My question: I would please appreciate help with a general proof of the assertion. (Also, while I have no real knowledge of Fourier series, is there a connection with this problem?)
Thanks
Integration by parts gives $$ \langle Df,g \rangle = \int_{-\pi}^{\pi} f'(x)g(x) \, dx = [f(x)g(x)]_{-\pi}^{\pi} + \int_{-\pi}^{\pi} -f(x)g'(x) \, dx. $$ All of the functions in $V$ have period $2\pi$, so the boundary terms vanish and the right-hand side becomes $ \langle f, -Dg \rangle $.