I'm currently studying a course on Advanced Real Analysis for a master degree, and our professor handed to everyone of us a 40-page book. I'm major in Algebra, so I'm not really comfortable with this subject. The book is so thin, and lack of information, and it's super hard to read. There's one part that I'm not quite understand, so I hope you guys'll help me with it.
Before posting here, I did do some resarch on the terminology, but there may be some part that I translate incorrectly, so if you spot out something wrong, just leave a comment below, and I'll fix it.
It's the proof on multiplication with smooth functions under the section Operations on Distributions. Ok, so here it goes:
We'll use the notation:
- $\mathcal{C}^\infty(\mathbb{R})$ for the vector space of all smooth functions on $\mathbb{R}$.
$\mathcal{D}(\mathbb{R})$ for the vector space of all smooth (infinitely differentiable), compactly support functions on $\mathbb{R}$
And on $\mathcal{D} (\mathbb{R})$, we define the norm of every $f \in \mathcal{D} (\mathbb{R})$ as follow: $||f||_N = \max\limits_{0 \le k \le N} \{ \max\limits_{x \in \mathbb{R}} \{ |f^{(k)}(x)|\} \}$
$\mathcal{D}_K$ for the sub-vector space of $\mathcal{D}(\mathbb{R})$ consisting of functions with support contains in $K$.
- $\mathcal{D}'(\mathbb{R})$ for the set of continuous linear functionals $\mathcal{D}(\mathbb{R})$
Multiplication with smooth functions
Let $F \in \mathcal{D}'(\mathbb{R})$, and $\varphi \in \mathcal{C}^\infty(\mathbb{R})$, we then define $\varphi F$ to be the following functional:
$$(\varphi F)(f) = F (\varphi f), \mbox{ for } f \in \mathcal{D} (\mathbb{R})$$
Now we'll prove that $\varphi F \in \mathcal{D}'(\mathbb{R})$. For all compact set $K \in \mathbb{R}$, we have: $$\exists C > 0, \exists N: |F(f)| \le C ||f||_N, \forall f \in \mathcal{D}_K$$
With $f \in \mathcal{D}_K$, we have:
$\left\{\begin{array}{l} \color{red}{\varphi f \in \mathcal{D}_K} \\ (\varphi f)^{(N)} = \sum\limits_{i = 0}^N C_N^i f^{(i)}\varphi^{N-i} \Rightarrow \color{red}{||\varphi f||_N \le C' ||f||_N} \end{array} \right.$
So $|(\varphi F) (f)| = |F(\varphi f)| \le CC' ||f||_N$.
The red parts are the parts that I don't understand, I don't really get why the support of $\varphi f$ should lie in $K$, it depends on how the $\varphi$ is defined right? What if $\varphi$ sends every 0 to some non-zero element, so the support of $\varphi f$ can span out of $K$?
And where do those iequalities come from?
And last but not least, can you guys recommend me any books (as elementary as possible) on this subject? As I don't think I can stand the book I'm currently reading.
The thing here is to prove that the linear functional $\varphi F: \mathcal D(\mathbb R) \to \mathbb R$ is continuous, ie, $\exists C > 0 $ such that $$ \vert \varphi F (f) \vert \leq C \Vert f \Vert_N, \ \forall f \in \mathcal D(\mathbb R). $$ The key here is that $\varphi$ is smooth and therefore all its derivatives up to $N$ are bounded on $K$ by a constant $C_\varphi$ (because $K$ is compact). So you can bound $\Vert (\varphi f)^{(N)} \Vert_N$ using the explicit expression, the constant $C'$ will depend on $C_\varphi$.
For the last inequality, it is a direct application of the previous result and the continuity of $F$.