Modern Abstract Analysis

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Martin Sleziak

11:52

Q: Unique functional $F$ on $\ell_q(\mathbb{N})$ extending $f$ and satisfying $\|F\|=\|f\|$

Let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$, $1< q<\infty$ and let $M=\{(a,b,c,0,\ldots):a,b,c\in \mathbb{F})$ be as subspace of $(\ell_q(\mathbb{N}),\|\cdot\|_q)$. Let $f:M\rightarrow\mathbb{F}$ by $(a,b,c,0,\ldots)\mapsto a-b-c$. Show that there exists a unique linear functional $F$...

We know that dual of $\ell_q$ can be identified with $\ell_p$. This means that $F$ has the form $F(x)=\sum y_nx_n$ for some $y\in\ell_p$. What can you tell about $y$ from the fact that $F|_M=f$? What do you get from the fact that $\|y\|_p=3^{1/p}$? Feel free to stop by in the functional analysis chatroom if more details are needed. — Martin Sleziak 1 min ago

1 hour later…

Jta

12:56

@MartinSleziak Okay so I guess we need $y=(1,-1,-1,y_4,\ldots)$? How can one conclude $\|y\|_p=3^{1/p}$?

Martin Sleziak

Well, you have $\|y\|_p=\sqrt[p]{\sum |y_k|^p}$.

In this case you have $\|y\|_p = \left(1^p+1^p+1^p + |x_4|^p + |x_5|^p + \dots \right)^{1/p}$.

So if any of the terms $x_4,x_5,\dots$ is non-zero, you get $\|y\|_p > (1+1+1)^{1/p} = 3^{1/p}$.

@Jta I do not know how experienced you are with chat, I' mention that to get MathJax rendered in chat you can use bookmarklet mentioned in this post on meta or go directly to robjohn's website

Perhaps I should have mentioned that you want $\|y\|=3^{1/p}$ because you are already know that $\|F\|=3^{1/p}$.

Jta

13:19

Ahh okay. Why does $\|F\|=3^{1/p}$ give $\|y\|=3^{1/p}$?

Is $y$ also an extension of $f$?

Martin Sleziak

What I am using here is that the dual of $\ell_q$ is isometric to $\ell_p$ using the natural identification.

Do you know that result?

I.e., $\ell_q^*\cong \ell_p$, where $1/p+1/q=1$.

And the isomorphism between $\ell_p$ and the dual of $\ell_q$ assigns to a function $y\in\ell_p$ the functional $$x\mapsto \sum_{n=1}^\infty x_ny_n.$$

@Jta Since $F\in\ell_q^*$, we know that $F$ is represented by a sequence $y\in\ell_p$ and that $\|F\|_{\ell_q^*}=\|y\|_{\ell_p}$.

BTW which textbook are you following?

Jta

Ahh thank you, that makes sense! I'm using Real Analysis by Folland

Martin Sleziak

If we have the same edition of the book, Folland shows in Section 6.2 that dual of $L_p(\mu)$ is isometrically isomorphic to $L_q(\mu)$.

I do not see whether he deals separately with the case of $\ell_p$ and $\ell_q$, but this can be considered a special case of the more general result (if $\mu$ is the counting measure on $\mathbb N$).

Martin Sleziak

14:03

@Jta If the above was sufficient for you to be able to solve the problem, perhaps you can post your attempt as an answer. In that way you have a chance that users will point out whether there are any mistakes or possible improvements.

And if something more needs to be said to help you with finishing the problem, feel free to ask here. (I usually look into this room when I notice that there is something new - and occasionally also some other users peek into this chatroom.)

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