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 $E\subseteq {\mathbb {R}}$ .

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 $\int _{E}f^{2},\int _{E}g^{2}$ 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.

Exercise 1. L² Sum Closure

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

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

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

Exercise 2. L² Is a Vector Space

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

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