MHB Problem of the Week #116 - August 18th, 2014

[Chris L T521]

Here's this week's problem!

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Problem: Let $V$ be the vector space of smooth (i.e. infinitely-differentiable) functions $f:\mathbb{R}\rightarrow\mathbb{C}$ such that for all $m,n\in\mathbb{N}$, there exists $C_{m,n}>0$ for which$$\forall\,x\in\mathbb{R},\,(1+x^2)^m\cdot|f^{(n)}(x)|\leq C_{m,n}.$$
This is called the Schwarz space - its elements are smooth functions all of whose derivatives decay rapidly. Define a function $P:V\times V\rightarrow\mathbb{C}$ by
$$P(f,g)=\int_{-\infty}^{\infty}f(x)g(x)\,dx.$$
Does $P$ define a pairing? Is $P$ nondegenerate?

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[Chris L T521]

This week's delays are brought to you by yours truly starting work again now that Fall semester is underway.

This week's problem was partially correctly answered by Ackbach. As for the case of whether or not the pairing is non-degenerate, I too am still working on that part and I hope to have a solution posted sometime in the next couple days.

Without much further ado, you can find Ackbach's partial solution below.

[sp]$P$ definitely defines a pairing. Check:
$\begin{align*}
P(r f,g)&=\int_{-\infty}^{\infty}(rf(x))g(x) \, dx \\
&=\int_{-\infty}^{\infty}f(x)(rg(x)) \, dx \\
&=P(f,rg) \\
&=r \int_{-\infty}^{\infty}f(x) g(x) \, dx \\
&=r P(f,g).
\end{align*}$
Moreover,
$\begin{align*}
P(f+g,h)&=\int_{-\infty}^{\infty}(f(x)+g(x))h(x) \, dx \\
&=\int_{-\infty}^{\infty}f(x) h(x) \, dx+\int_{-\infty}^{\infty}g(x) h(x) \, dx \\
&=P(f,h)+P(g,h),
\end{align*}$
and similarly, $P(f,g+h)=P(f,g)+P(f,h)$. Thus, $P$ satisfies all the requirements for a pairing.[/sp]