MHB Problem of the Week #32 - November 5th, 2012

[Chris L T521]

Thanks to those who participated in last week's POTW! Here's this week's problem!

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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 $f\in V$).

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[Chris L T521]

This week's question was correctly answered by Sudharaka. You can find his solution below.

Spoiler

\begin{eqnarray}

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

&=&\int_0^{2\pi}\overline{f(\theta) \overline{g(\theta)}} \,d\theta\\

\end{eqnarray}

For a complex valued function of a real variable, \(\int_{a}^{b}\overline{f(x)}\,dx=\overline{\int_{a}^{b}f(x)\,dx}\). Therefore,

\begin{eqnarray}

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

&=&\overline{\langle g,f\rangle}

\end{eqnarray}

\begin{eqnarray}

\langle f,f\rangle&=&\int_0^{2\pi}\overline{f( \theta)}f(\theta) \,d\theta\\

\end{eqnarray}

Let, \(f(\theta)=f_{1}(\theta)+if_{2}(\theta)\) where \(f_{1}\) and \(f_{2}\) are real valued functions. Then we get,

\[\langle f,f\rangle=\int_0^{2\pi}[f_{1}(\theta)]^{2} \,d\theta+\int_0^{2\pi}[f_{2}( \theta)]^{2}\,d\theta\]

\[\therefore\langle f,f\rangle>0\mbox{ for any non zero function }f\in V\]