0
$\begingroup$

Context

I read this post which asks for an explicit norm on $C^0\left(\Bbb R,\Bbb R\right)$.

Obviously, usual norms don't work because of the fact $\Bbb R$ is infinite so using an integral would fail even on constant functions and $\sup$ would fail on affine functions.

The integral looking too difficult to fix for this space, I decided to try to fix the $\sup$ by crushing functions when they went to $\pm\infty$.

To do that, I assumed there was a sequence of functions $\left(f_n\right)_{n\in\mathbb{N}}\in \mathcal C^0\left(\Bbb R,\Bbb R\right)^n$ so that $\forall g \in C^0\left(\Bbb R,\Bbb R\right)^n,\exists n \in \Bbb{N}, \cfrac{g}{f_n}$ is bounded on $\Bbb{R}$. And I defined the norm to be $\|g\|=\sup \left|\cfrac{g}{f_n}\right|$ for the smallest such $n$.

Homogeneity and separation worked well but subadditivity did't work so I favored the post to have a look later to see if someone else answered and gave up.

And since then, I'm wondering whether such a sequence of functions exists.


The question

Is there a sequence of functions $\left(f_n\right)_{n\in\mathbb{N}}\in \mathcal C^0\left(\Bbb R,\Bbb R\right)^n$ so that $\forall g \in C^0\left(\Bbb R,\Bbb R\right)^n,\exists n \in \Bbb{N}, \cfrac{g}{f_n}$ is bounded on $\Bbb{R}$?


Thoughts and comments

My first thought was to take something like $f_n=exp^n=\exp \circ \dots \circ \exp$ but I have no idea how to prove or disprove that it works... At sure I was fairly convinced it would work...

But then I thought of taking something like $f_n(x)= x^n$. Of course this doesn't work since if you took $g=\exp$, $\cfrac{g}{f_n}$ wouldn't be bounded for any $n$. But a few years ago, I would've thought it would work...

Then I tried to search for functions for which $f_n=\exp^n$ wouldn't work. And the only thing I could think of was Ackermann's idea of "creat[ing] an endless procession of arithmetic operations" (which I discovered while skim-reading this). The same way you go from $x\mapsto x+a$ to $x\mapsto ax$ by making the number of $a$s you add depend on $x$, we then have $x\mapsto a^x$ where the number of times you multiply by $a$ depends on $x$ so I guess we could apply the same kind of method once again that would make the $\exp^n$ fail as my $f_n$ but I can't conceive such functions. I'm not even sure they are possible to define...


Thoughts and comments v2

$f_0$ defined by:

$f_0(0)=1$

$\forall n \in\Bbb{N}^*,f_0(n)=f_0(-n)=$ $n^{th}$ Ackermann number

And then the rest is just lines between those previously defined points.

And then $\forall n \in\Bbb{N}^*,f_{n+1}=f_n\circ f_0$


Thank you in advance for your answers. :)

$\endgroup$
2
$\begingroup$

Let $g$ interpolate the values $g(n)=n \max_{k<n} |f_k(n)|$. Then $\left|\frac {g(x)}{f_n(x)}\right|\ge m$ for $x=m$, hence the quotient is not bounded.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.