Measure Theory/Linfinity

The Relationship between Various Lp

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There is just one loose-end that we left dangling a couple lessons before, which is the issue of when  . There we asked whether it was possible for to make sense of the idea that the conjugate of p is  , but for this to make sense we need to have some notion of   and  .

It is also natural to wonder if there is a relationship between   and   if  . Indeed there is a simple relationship.

Moreover, if we let   then do we discover a notion of   which perhaps "plays nicely" with  ?


Exercise 1. A Strict Separation of   Spaces


Let  . Show that  .

Use this basic idea to show that for any   there is always a function in

 

In something like a converse, also show that there is always a function in

 

The above exercise shows that, if there is to be some relationship between   and  , it will not hold generally -- and it may depend on the nature of the subset E.


Exercise 2.  


1. Let   and show that there exists some constant   (which depends on p and q) such that for every  ,

   

Hint: Starting from the left side, write down a factor of 1 next to   and apply the product bound. It may seem natural, when using the product bound, to take the p norm of   but in fact this fails to produce any nice "cancellations".

Instead, prove that   and work through the remaining details.

2. Now argue that  .  Note that the containment is strict so you have two things to show: Both that the subset relation holds, and that there is some element in the set difference.
3. Finally, show how the above result can be generalized from the interval   to some wider class of subsets of real numbers.  At what point did you use any facts about  , and can you remove some details about this set which were unnecessary for the above proof?

L

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As in the previous exercise, we will assume that  . (This is the generalization that you should have found at the end.

We now have some sense of what happens as we let p in   grow larger. In particular, you get an ever shrinking sequence of spaces of functions.

What then is the set

 

This is essentially the same question as finding a characterization of  .

Definition:  

Let   be a subset of real numbers and   for every  . Then define the infinity norm by

 

and the space of functions with finite infinity norm,

 


Exercise 3. Experimenting with  
1. Set  .  Compute  .  

Hint: If you're a little rusty with your tricky limits at infinity, use the "e to the ln" trick.

2. Set   for some  .  Compute  .
3. Set  .  Compute  .
4. What happens if you multiply any of these functions, in any of these examples, by a constant?  What happens if you sum any of these with a constant?

Hint: Use the homogeneity of the norm for the constant multiple part -- don't make things harder than they have to be.

For the constant sum part, if you consider   it will be helpful to consider the following:

Let  . Then

 
5. Based on all of the experimentation above, what do all of these values have in common?  

The above exercise hopefully suggests that   is the maximum of the function ... er, well, maybe not exactly the maximum of the function. After all, the very well-behaved function   on the interval (0,1) has no maximum.

Ok, so maybe   is really the supremum rather than the maximum. Well, but even this doesn't work, because on a null set we can mess around with the function values and not change the value of the   norm.

Therefore we need a definition like this but which is not sensitive to values on any null set. Rather than talking about a maximum, or a supremum, we talk about an "essential supremum".

Definition: essential bound, essential supremum

Let   be any function defined on a subset of real numbers. Let  .

We say that C is an essential bound of f if there exists some null set   such that   for every  .

If   is the set of all essential bounds of f,

 

then we define the essential supremum of f by the infimum of  . That is to say, the essential supremum of f is

 


Exercise 4.  


Let   and let   have a finite essential supremum, which we denote by  .

1. Prove that  .

This will require using some analysis of integrals that we haven't done for a while. But it is natural to consider the indexed family of sets

  for any  

With this one may apply the ML bound to obtain the desired result.

2. Prove  .

For the reverse inequality, let   be arbitrarily small. Let

 

Now prove that

 

Then take the limits   and then  .


Exercise 5. Further Extending  


Show that the notion of an   space can be extended to include  . That is to say

1. Define the norm, space, and distance for  .

2. Prove that it is a vector space.

3. Prove that the concept of the conjugate of a real number   can be extended to include  , and that the product-to-sum bound still applies.

4. Prove that the integral product bound does not hold but the triangle inequality still holds when  , and that   is complete. Be careful about any conditions which must hold in order for theorems to hold -- they may not be exactly the same conditions for finite p.