Measure Theory/L2 Vector Space

Closure Properties

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One of the most fundamental properties one can ask for when investigating a set equipped with some operations, is: Is the set closed?

Here we have the set   which, recall, is the set of all functions with finite   norm, and  .

We start by considering the operation of summing two functions. Then we would like to show that if   then  .

By definition, this means that we want to prove, if  , then

 

This, in turn, means that by the finiteness of   we would like to prove that   is finite.

A natural instinct is to take the max,   which we know from earlier work is integrable -- but it is not clear that it is "square integrable". We'll need to try something else.


Exercise 1. L2 Sum Closure


Inspired by the above, with  , show that  . Moreover, and by a similar logic,  .

Then use this to show that  .

Then use this result to infer the closure of   under addition.


Exercise 2. L2 Is a Vector Space


Show that   is a vector space over  . Recall the vector space properties:

1. Closure under sums and scalar multiples.
2. Associativity, commutativity, identity, and closure under inverses, for vector addition.  
3. Associativity and identity for scalar multiplication.
4. Scalar and vector distribution.