I found this problem on a teacher's handout:
Prove that the decomposition into even and odd functions over $ C^0([-a,a],\mathbb{C}) $ is orthogonal if we use the inner product: $$ \langle f,g \rangle = \int^a_{-a} f(t) \overline{g(t)} dt $$
I'm having trouble understanding what this means:
$ C^0([-a,a],\mathbb{C}) $
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Edit: Also, what does the zero superscript mean?
In general, how do I use that notation for any other vector spaces?
Is the expression before the comma a domain? Is the $\mathbb{C}$ after the comma a field?
Stripped of the verbiage, the problem is simply to show that if $f$ is odd and $g$ is even, or vice-versa, then $\int_{a}^{a}{f(t)\overline{g(t)} \text{dt}}=0$
The complex conjugate of an even function is even, and likewise, the conjugate of an odd function is odd, so in any event, the integrand is the product of an odd function an an even function. Now the product of an odd function and an even function is odd, for suppose $h$ is even and $k$ is odd. Then $$hk(-z)=h(-z)k(-z) = h(z)(-k(z))=-h(z)k(z)=-hk(z)$$
Since the integrand is odd, the integral is $0$.