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 $E\subseteq ℝ$.

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 $\Vert f{\Vert }_2,\Vert g{\Vert }_2<\mathrm{\infty }$, then

$\Vert f+g{\Vert }_2<\mathrm{\infty }$

This, in turn, means that by the finiteness of $\underset{E}{\int }{f}^2,\underset{E}{\int }{g}^2$ we would like to prove that $\underset{E}{\int }(f+g{)}^2$ 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 $ℝ$. 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.