- #1
looserlama
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Homework Statement
For F [itex]\in[/itex] {R,C} and for an infinitie discrete time-domain T, show that lp(T;F) is a strict subspace of c0(T;F) for each p [itex]\in[/itex] [1,∞). Does there exist f [itex]\in[/itex] c0(T;F) such that f [itex]\notin[/itex] lp(T;F) for every p [itex]\in[/itex] [1,∞)
Homework Equations
Well we know from class that lp(T;F) = {f [itex]\in[/itex] FT | Ʃ |f(t)|p < ∞} for p [itex]\in[/itex] [1,∞) and c0(T;F) = {f [itex]\in[/itex] FT | [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.
The Attempt at a Solution
Well as I said above, we know that both lp(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 [itex]\in[/itex] lp(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] c0(T;F)
Therefore lp(T;F) [itex]\subseteq[/itex] c0(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] FT by f(t) = [itex]\frac{1}{t1/p}[/itex] if t ≠ 0 and 0 if t = 0.
Clearly f [itex]\in[/itex] c0(T;F) as it's limit goes to 0.
And it's easy to show that f is not in any lp(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 lp(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 c0 but not in lp for EVERY p [itex]\in[/itex] [1,∞).
Any help would be awesome!