Measure Theory/L2 Vector Space
Closure Properties
editOne 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
Then use this to show that . Then use this result to infer the closure of under addition. |
Exercise 2. L2 Is a Vector Space
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. |