Articles of real analysis

Maximum principle for subharmonic functions

Let $\Omega$ be a domain of $\mathbb{R}^n$, and $u:\Omega\to\mathbb{R}$ a continuous function. We call $u$ subharmonic if for any ball $B\subset\subset\Omega$ and any $h:\overline B\to\mathbb{R}$ which is continuous on $\overline B$, harmonic in $B$, and satisfies $u|_{\partial B}\leq h|_{\partial B}$, then $u\leq h$ on the whole $\overline B$. No regularity assumed for $u$, except for […]

Greatest common divisor of real analytic functions

Consider two real-valued real analytic functions $f$ and $g$. I want to prove that there exists a greatest common divisor $d$, which is a real analytic function. By greatest common divisor, I mean the following: Common divisor: There exist real analytic functions $q_1, q_2$ such that $f = dq_1, g = dq_2$, and Greateast: If […]

$\lim_{p\rightarrow\infty}||x||_p = ||x||_\infty$ given $||x||_\infty = max(|x_1|,|x_2|)$

I have seen the proof done different ways, but none using the norm definitions provided. Given: $||x||_p = (|x_1|^p+|x_2|^p)^{1/p}$ and $||x||_\infty = max(|x_1|,|x_2|)$ Prove: $\lim_{p\rightarrow\infty}\|x\|_p = \|x\|_\infty$ I have looked at the similar questions: The $ l^{\infty} $-norm is equal to the limit of the $ l^{p} $-norms. and Limit of $\|x\|_p$ as $p\rightarrow\infty$ but […]

if $A, B$ are open in $\mathbb R$ then so is $A+B.$

I am trying to find out a counterexample to the problem: if $A, B$ are open in $\mathbb R$ then so is $A+B.$ But I could not find any such counterexample. Please help me.

Show that $\int_0^\infty\frac{1}{x ((\ln x)^2+1)^p}dx$ converges for any $p\geq 1$ and find its value.

Suppose that $p\geq 1.$ In this question, Robert answered that the following integral $$\int_0^\infty \frac{1}{x ((\ln x)^2+1)^p} dx$$ converges for any $p\geq 1.$ However, I am not able to show Robert’s claim. Below is my attempt: Use the substitution $u = \ln (x).$ Then we have $$\int_0^\infty \frac{1}{x ((\ln x)^2+1)^p} dx = \int_{-\infty}^\infty \frac{1}{(u^2+1)^p} du.$$ […]

Theorem 3.55 Rudin (rearrangement and convergence)

If $\sum a_n$ is a series of complex numbers which converges absolutely, then every rearrangement of $\sum a_n$ converges, and they all converge to the same sum. Proof: Let $\sum a_n’$ be a rearrangement , with partial sums $s_n’$. Given $\epsilon > 0$ there exist an integer $N$ such that $m \geq m \geq N$ […]

$f()=$ for continuous strictly monotonic function

Let $f$ is continuous and strictly monotonic function on $[a,b]$. Prove that $$f([a,b])=[f(a),f(b)].$$ Proof: We know that $[a,b]$ is a connected set in $\mathbb{R}$ then $f([a,b])$ is connected since $f$ is continuous. Also $f(a),f(b)\in f([a,b])$. Hence $[f(a),f(b)]\subseteq f([a,b]).$ Let $y\in f([a,b])$ then $y=f(x)$ for some $x\in[a,b]$. Since $x\in[a,b]$ then $f(a)\leqslant y\leqslant f(b)$. Hence $f([a,b])\subseteq [f(a),f(b)]$ […]

An equality concerning the Lebesgue integral

Let $f:X\mapsto[0,+\infty)$ be a non-negative measurable function defined on the space $X$, endowed with the complete $\sigma$-additive, $\sigma$-finite, measure $\mu$ defined on the $\sigma$-algebra of the measurable subsets of $X$. I have read that the following equality holds for the Lebesgue integral: $$\int_X f d\mu = \int_{[0,+\infty)} \mu(\{x\in X: f(x)>t\}) d\mu_t$$where $\mu_t$ is the usual […]

Proof of $\limsup\sin nx=1, n\rightarrow \infty \forall x\in \mathbb{R}.$

I have a problem: need to prove $$\limsup_{n\to\infty}\sin nx=1$$ for all x without set which measure is equal zero. We need to come up with a sequence with limit that equals to one, but I don’t know how to do this.

Show that if $x,x^{'}\in C$ then $C_x=C_{x^{'}}$.

Let $C$ be a closed convex set in $\Bbb R^2$. Define $C_x=\{y:x+ty\in C\forall t>0\}$. Show that if $x,x^{‘}\in C$ then $C_x=C_{x^{‘}}$. Attempt: Since $x,x^{‘}\in C$ we have a continuous path $f:[0,1]\to C$ such that $f(0)=x,f(1)=x^{‘}$ Let $y_1\in C_x\implies x+ty_1\in C$. To show that $x^{‘}+ty_1\in C_{x^{‘}}$. I need a path in $C$ which joins $x^{‘}+ty_1$ with […]