The Schwartz space $\mathcal{S}(\Bbb{R})$ consists of all indefinitely differentiable functions $f$ such that for all $\ell, k \geq 0$, we have $$ \sup_{x \in \Bbb{R}} |x|^k |f^{(\ell)}(x)| \lt \infty $$ i.e. $f$ and all of its derivatives are rapidly decreasing in that sense. How do I show that this is true of $f(x) = e^{-x^2}$? […]

Let $f\in\mathscr{S}(\mathbb R^n)$ where $\mathscr{S}(\mathbb R^n)$ is the Schwartz space. Is it true that $\partial^\alpha f$ is a uniformly continuous function for all multi-index $\alpha$? Here $\mathscr{S}(\mathbb R^n)$ is the $\mathbb C$-vector space of all the $C^\infty$ functions $f:\mathbb R^n\longrightarrow \mathbb C$ for which $$x\longmapsto x^\alpha \partial^\beta f(x),$$ is a bounded function in $\mathbb R^n$ […]

*****Note: Parts A, C and D I managed. Only need help on part B now would really would appreciate the help on B Hi, in my summer real analysis (or measures and real analysis as my instructor refers to it) I was presented with this question from Folland’s real analysis second edition on distribution theory […]

Let $X$ be a set, $\mathcal F$ a $\sigma$-field of subsets of $X$, and $\mu$ a probability measure on $X$. Given random variables $f,g\colon X\rightarrow\mathbb{R}$ such that $$\int_\mathbb{R}hd{\mu_f}=\int_{\mathbb{R}}hd\mu_g$$ for any Schwartz function $h$. Is it necessarily true that $\mu_f=\mu_g$?

Everybody says that the Schwartz space is a space of rapidly decreasing functions, or of functions that rapidly vanish, but I am baffled with proving it formally. I can come up with nice reasons why it is true, but not with a solid formal proof. What I want to prove is $$f\in S(\mathbb{R}^n)\implies\lim_{|x|\to\infty}f(x)=0$$ or more […]

Let $f\in S_\infty\subset L_1(\mathbb{R},\mu)$ with $\mu$ as the Lebesgue linear measure be a Lebesgue-summable function such that $$\forall (p,q)\in\mathbb{N}^2_{\ge 0}\quad\exists C_{pq}>0: \Bigg|x^p\frac{d^q}{d x^q}f(x)\Bigg|< C_{pq}$$ I was wondering whether, if $\forall p\in\mathbb{N_{\ge 0}}\quad\int_{\mathbb{R}}x^pf(x)d\mu=0$, then $f$ is constantly, or almost everywhere, null. I cannot find a counterexample and therefore I think that the implication might well hold, […]

I’m wondering about some of the topological properties of $\mathcal D(\Omega)$ and $\mathcal D'(\Omega)$: I know $\mathcal D(\Omega)$ is not metrizable, so not first countable (right?). However, my question is: Is $\mathcal D(\Omega)$ sequential? When $\mathcal D'(\Omega)$ is endowed with the weak* topology, is it sequential? (I presume this one is clearly not first countable […]

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