- Thread starter
- Moderator
- #1

- Jan 26, 2012

- 995

-----

**Problem**: Let $V$ be the space of differentiable complex-valued functions on the unit circle in the complex plane, and for each $f,g\in V$, define

\[\langle f,g\rangle=\int_0^{2\pi}\overline{f(\theta)} g(\theta) \,d\theta.\]

Show that this form is Hermitian (i.e. $\langle f,g\rangle = \overline{\langle g,f\rangle}$) and positive definite (i.e. $\langle f,f\rangle > 0$ for all nonzero functions $f\in V$).

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!