Prove l^p strict subspace of c0

[looserlama]

Homework Statement

For F \in {R,C} and for an infinitie discrete time-domain T, show that l^p(T;F) is a strict subspace of c₀(T;F) for each p \in [1,∞). Does there exist f \in c₀(T;F) such that f \notin l^p(T;F) for every p \in [1,∞)

Homework Equations

Well we know from class that l^p(T;F) = {f \in F^T | Ʃ |f(t)|^p < ∞} for p \in [1,∞) and c₀(T;F) = {f \in F^T | \forallε \in R_>0, \exists a finite set S \subseteq T such that {t \in T | |f(t)| > ε} \subseteq S} (\Leftrightarrow t \rightarrow ±∞ \Rightarrow f(t) \rightarrow 0) are both vector spaces.

The Attempt at a Solution

Well as I said above, we know that both l^p(T;F) and c₀(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 \in l^p(T;F) \Rightarrow Ʃ|f(t)|^p < ∞
Therefore lim as |t|\rightarrow∞ of |f(t)|^p = 0 \Rightarrow lim as |t|\rightarrow∞ of |f(t)| = 0 \Rightarrow f \in c₀(T;F)

Therefore l^p(T;F) \subseteq c₀(T;F).

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

This is what I though of:

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

Clearly f \in c₀(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 \in [1,∞).

But to properly do this problem I need to find a function that's not in l^p(T;F) for every p \in [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 c₀ but not in l^p for EVERY p \in [1,∞).

Any help would be awesome!

 

[micromass]

Think about using logarithms.

 

[looserlama]

Ok.

That makes sense as ln(t) increase much slower than t.

So this is what I was thinking:

Let f(t) = \frac{1}{ln(|t| + 1)} if t ≠ 0 and 0 if t = 0.

So clearly lim as |t|→∞ of f(t) = 0, so it is in c₀.

But then showing it isn't in l^p for every p is a bit harder.

This is how I tried:

Basically I wanted to use the comparison test to show Ʃ|\frac{1}{ln(|1| + 1)}|^p = Ʃ1/|ln(|t| + 1)|^p≤ Ʃ\frac{1}{|t|} and since that diverges \Rightarrow Ʃ|f(t)|^p diverges.

The problem is showing that \exists N \in T such that \forall t ≥ N |ln(|t| + 1)|^p ≤ |t|.

I tried differentiating both sides but it ends up giving p|ln(|t| + 1)|^{p - 1} ≤ |t| +1 which is essentially the same thing as before.

It makes sense to me as ln(t) increases much slower than t, and we can always find an N for which ln(t)^p will be less than t for any t ≥ N. But I don't know how to show that.

I also thought of using the ratio or root test, but that seems like it wouldn't work very well...

Any thoughts?