Articles of analysis

Is the ring of germs of $C^\infty$ functions at $0$ Noetherian?

I’m considering the property of the ring $R:=C^\infty(\mathbb R)/I$, where $I$ is the ideal of all smooth functions that vanish at a neighborhood of $0$. I find that $R$ is a local ring of which the maximal ideal is exactly $(x)$. I also want to determine if $R$ is a Noetherian ring, but I have […]

$\lim_{x\to 0+}\ln(x)\cdot x = 0$ by boundedness of $\ln(x)\cdot x$

I saw a proof that $$ \lim_{x\to 0} \ln|x|\cdot x = 0 $$ where is is argued that for $x \in (0,1)$ we have $$ | \ln(x) x | = \left| \int_1^x x/t ~\mathrm d t \right| = \left| \int_x^1 x/t ~\mathrm d t \right| \le \left|\int_x^1 1 dt\right| = |1 – x| \le 1 […]

Application of Weierstrass theorem

Let f be a continuously differentiable function on $[a.b]$. Show that there is a sequence of polynomials $\{P_n\}$ such that $P_n(x) \to f$ and $P’_n(x) \to f’ (x)$ uniformly on $[a,b]$ My approach has been as follows. Since f is continuously differentiable, we have $Q_n(x) \ to f'(x)$ on$ [a,b]$ uniformly (Weierstrass) . I’m not […]

Doubling measure is absolutely continuous with respect to Lebesgue

Let $\mu$ be a fixed finite measure on $\mathbb R$. We say that $\mu$ is doubling if there exists a constant $C>0$, such that for any two adjacent intervals $I=[x−h,x]$ and $J=[x,x+h]$, $$C^{−1}\mu(I)≤\mu(J)≤C\mu(I).$$ Assuming that $\mu$ is doubling, show that there exist positive constants $B$ and $a$, such that for every interval $I$, $$\mu(I)≤B[length(I)]^a$$ By […]

Compute $\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$

Compute $$\int_0^{\infty}\frac{\cos(\pi t/2)}{1-t^2}dt$$ The answer is $\pi/2$. The discontinuities at $\pm1$ are removable since the limit exists at those points.

Proof involving norm of an integral

I am totally stuck and have no idea whatsoever on how to prove the following inequality (by the way this is a problem from an undergraduate book in multivariable advanced calculus at Junior/Senior level ): Let $g=\left ( g_{1},g_{2},…,g_{n} \right ): \left [ a,b \right ]\rightarrow \mathbb{R}^{n}$ is a continuous function, then we define: $\int_{a}^{b}g\left […]

$X, Y \subset \mathbb{R}^n$. Define $d(K, F) = \inf\{ d(x,y), x \in K, y \in F\}$, show that $d(a,b) = d(K,F)$ for some $a$, $b$

Some caveats: Let $K$ be non-empty and compact, $F$ be non-empty and closed, $X, Y \subset \mathbb{R}^n$. Define $d(K, F) = \inf\{ d(x,y), x \in K, y \in F\}$, where $d(x,y)$ is one of the metrics $d_1$, $d_2$, or $d_\infty$ on $\mathbb{R}^n$. Show that $d(a,b) = d(K,F)$ for some $a\in K$, $b\in F$. It seems […]

Problem about $G_{\delta}$-set and $F_{\sigma}$-set

Prove if $E$ is any measurable subset of $\mathbb{R}$, then there are a $G_{\delta}$-set $G$ and a $F_{\sigma}$-set $H$ such that $H \subseteq E \subseteq G$, and such that $m(G$\ $H)=0$. In order to prove this, for each $n \in \mathbb{Z}$ let $E_n=E \cap (n,n)$. How to show that the regularity theorem for $E_n$ can […]

Riemann-Stieltjes integral of unbounded function

In many theorems about the Riemann-Stieltjes integral they required the hypothesis of $f$ to be bounded (for example: Suppose that $f$ is bounded in $[a,b]$, $f$ has only finitely many points of discontinuity in $I=[a,b]$, and that the monotonically increasing function $\alpha$ is continuous at each point of discontinuity of $f$, then $f$ is Riemann-Stieltjes […]

Show that for any $g \in L_{p'}(E)$, where $p'$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$

Let $1 < p < \infty, f_k \in L_p(E), k = 1, 2, …, $ and $\lim_{k \rightarrow \infty}f_k(x) = f(x)$ a.e., $\sup_{1 \leq k<\infty}||f_k||_p \leq M$. Show that for any $g \in L_{p’}(E)$, where $p’$ is the conjugate of $p$, $\lim_{k \rightarrow \infty}\int_Ef_k(x)g(x)dx = \int_Ef(x)g(x)dx$ This is from a past qual. Not really sure […]