Measure Theory/L2 Vector Space

Closure Properties
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 $$L^2(E)$$ which, recall, is the set of all functions with finite $$L^2(E)$$ norm, and $$E\subseteq \Bbb R$$.

We start by considering the operation of summing two functions. Then we would like to show that if $$ f,g \in L^2(E)$$ then $$f+g\in L^2(E)$$.

By definition, this means that we want to prove, if $$\|f\|_2,\|g\|_2<\infty$$, then


 * $$\|f+g\|_2<\infty$$

This, in turn, means that by the finiteness of $$ \int_Ef^2,\int_Eg^2$$ we would like to prove that $$\int_E(f+g)^2$$ is finite.

A natural instinct is to take the max, $$h=\max\{f,g\}$$ 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.

Inspired by the above, with $$h=\max\{f,g\}$$, show that $$f,g\le h \le |f|+|g|$$. Moreover, and by a similar logic, $$h^2\le f^2+g^2$$.

Then use this to show that $$(f+g)^2 \le (2h)^2\le 4(f^2+g^2)$$.

Then use this result to infer the closure of $$L^2(E)$$ under addition.

Show that $$L^2(E)$$ is a vector space over $$\Bbb R$$. 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.