(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For F [itex]\in[/itex] {R,C} and for an infinitie discrete time-domain T, show that l^{p}(T;F) is a strict subspace of c_{0}(T;F) for each p [itex]\in[/itex] [1,∞). Does there exist f [itex]\in[/itex] c_{0}(T;F) such that f [itex]\notin[/itex] l^{p}(T;F) for every p [itex]\in[/itex] [1,∞)

2. Relevant equations

Well we know from class that l^{p}(T;F) = {f [itex]\in[/itex] F^{T}| Ʃ |f(t)|^{p}< ∞} for p [itex]\in[/itex] [1,∞) and c_{0}(T;F) = {f [itex]\in[/itex] F^{T}| [itex]\forall[/itex]ε [itex]\in[/itex] R_{>0}, [itex]\exists[/itex] a finite set S [itex]\subseteq[/itex] T such that {t [itex]\in[/itex] T | |f(t)| > ε} [itex]\subseteq[/itex] S} ([itex]\Leftrightarrow[/itex] t [itex]\rightarrow[/itex] ±∞ [itex]\Rightarrow[/itex] f(t) [itex]\rightarrow[/itex] 0) are both vector spaces.

3. The attempt at a solution

Well as I said above, we know that both l^{p}(T;F) and c_{0}(T;F) are vector spaces, so to show that one is a subspace of the other it is sufficient to show that one is a subset of the other.

So,

Let f [itex]\in[/itex] l^{p}(T;F) [itex]\Rightarrow[/itex] Ʃ|f(t)|^{p}< ∞

Therefore lim as |t|[itex]\rightarrow[/itex]∞ of |f(t)|^{p}= 0 [itex]\Rightarrow[/itex] lim as |t|[itex]\rightarrow[/itex]∞ of |f(t)| = 0 [itex]\Rightarrow[/itex] f [itex]\in[/itex] c_{0}(T;F)

Therefore l^{p}(T;F) [itex]\subseteq[/itex] c_{0}(T;F).

Now this is the hard part, showing that it is a strict subset.

This is what I though of:

Define f [itex]\in[/itex] F^{T}by f(t) = [itex]\frac{1}{t^{1/p}}[/itex] if t ≠ 0 and 0 if t = 0.

Clearly f [itex]\in[/itex] c_{0}(T;F) as it's limit goes to 0.

And it's easy to show that f is not in any l^{p}(T;F) for any specific p [itex]\in[/itex] [1,∞).

But to properly do this problem I need to find a function that's not in l^{p}(T;F) for every p [itex]\in[/itex] [1,∞).

i.e., my problem is, I have to chose a p first, then this works, but whatever p I chose, f will not be in it's space, but it will be in the p+1 space. So it doesn't work for for every p, only a specific p.

So pretty much I can't think of a function that would be in c_{0}but not in l^{p}for EVERY p [itex]\in[/itex] [1,∞).

Any help would be awesome!

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# Homework Help: Prove l^p strict subspace of c0

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