Quantcast
Channel: Active questions tagged real-analysis - Mathematics Stack Exchange
Browsing all 9124 articles
Browse latest View live
↧

If $\lim_{n \to \infty} n x_n = 0$ then $\lim_{n\to \infty} (1+x_n)^n =1$

The following appears without proof in some lecture notes about exponents (specifically, it was used to prove the limit $\lim_{n\to \infty} (1+\frac{x}{n})^n = e^x := \sup\{ e^q \mid q\in \Bbb Q, q\le...

View Article


Extension of finitely additive function to a countably additive measure

Let X be an infinite set. Let $\mathcal{A}$ be the collection of subsets A of X such that either A is finite, and then m(A):=0, or the complement of A is finite, and then m(A):=1.Under what condition...

View Article


Boundary properties of the convex envelope

Let$\Omega \subset \mathbb{R}^n$ be open, bounded and strictly convex, i.e. for all$x, y \in \overline{\Omega}$and$\lambda \in (0, 1)$we have that$$ \lambda x + (1 - \lambda)y \in \Omega.$$Furhermore,...

View Article

Real Analysis , Folland Problem 6.1.5

Problem 6.1.5 - Suppose $0 < p < q < \infty$. Then $L^p \not\subset L^q$ if and only if $X$ contains sets of arbitrary small positive measure, and $L^q\not\subset L^p$ if and only if $X$...

View Article

Question regarding Lebesgue Integrability in $\sigma$ -finite spaces

I'm taking a course in measure theory and we defined integrability in a $\sigma$ -finite space as follows: Suppose $\left(X,\mathcal{F},\mu\right)$ is a $\sigma$-finite measure space, a measurable...

View Article


$\lim_{\epsilon \rightarrow 0} \frac 1{\epsilon} \int_0 ^{\infty} f(x...

Check if the following limit exist and find the value if it exist .$\lim_{\epsilon \rightarrow 0} \frac 1{\epsilon}\int_0 ^{\infty} e^{-x/{\epsilon} }(\cos (3x)+ x^2 +\sqrt{x+4} ) dx $My Attempt :-For...

View Article

Example for a non-separable space of compactly supported continuous functions

For a topological space $X$ let $C_c(X)$ be the space of compactly supported continuous real-valued functions. It is well known that $C_c(X)$ (provided with the sup norm) is separable, if $X$ is a...

View Article

Is Rudin's proof of the inverse function theorem correct?

This question concerns the proof of the inverse function theorem found in Walter Rudin's Principles of Mathematical Analysis (3rd ed.).It defines for each $y \in...

View Article


Dose $\lim_{k\to\infty}\langle f_k\rangle=\langle f\rangle$ imply...

We know that for any $\langle f\rangle\in L^p(X,\mathscr{A},\mu)$ there is a sequence $\{\langle f_k\rangle\}$ of elements of $L^p(X,\mathscr{A},\mu)$, where each $f_k$ is a simple function that...

View Article


Subset of $\mathbb{R}$ that satisfies trichotomy and multiplicative closure...

Is it possible to construct a subset $\mathbb{P}$ of $\mathbb{R}$ thatsatisfies trichotomy and multiplicative closure but not additive closure?Trichotomy says that if $x\in\mathbb{R},$ then exactly one...

View Article

Showing a $g\in L^{q}([0,1])$ and proving an inequality for its $q$-norm

I'm working through some problems dealing with $L^p$ spaces and came across this specific exercise:Let $1<p, q<\infty$ be such that $\frac{1}{p}+\frac{1}{q}=1$. For any $f \in L^{p}([0,1])$,...

View Article

Image of an increasing function with discontinuities on a densely ordered...

When discussing this question, we made interesting observations on increasing functions with discontinuities on rationals. Let $f: [0,1] \to [0,1]$ be an increasing function with discontinuities on...

View Article

Finding a fit function with specific properties

I am looking for a function f(x) for x>0 to fit to monte carlo simulation results. Here is an example of a simulated family of curves that I wish to fit:simulation resultsFrom the physical process...

View Article


Well-posedness for system of linear PDES

I am studying systems of partial differential equations (PDEs) with smooth coefficients and am trying to understand the general solution. Consider the following system:\begin{align}u_x &=...

View Article

Pointwise convergence of almost periodic function

This question is inspired by this post and definitions I use can be found there.Fix $f$ a real-valued almost periodic (but not periodic) function. For each $t\in\mathbb{R}$, define $f_t(x) = f(x-t)$...

View Article


$f \in C^{\infty} (\mathbb{R}^3, \mathbb{R}^4)$,...

We have $f \in C^{\infty} (\mathbb{R}^3, \mathbb{R}^4)$, $f = f_1, f_2, f_3, f_4$.Elements of that transformation satisfy:$$f_1(x)^2 + f_2(x)^2 = 1 = f_3(x)^2 + f_4(x)^2 \ \text{ for: } \ x \in...

View Article

$\omega = \frac{x(x^2+y^2-1)}{(x^2+y^2+1)^2-4x^2}dy+...

I found such problem with solution.Problem:We have:$C1 = \{ (x, y) : (x-1)^2 + y^2 = \frac{1}{4} \}$$C2 = \{ (x, y) : x^2 +y^2 = 4 \}$$C3 = \{ (x, y) : (x + 1)^2 + y^2 = \frac{1}{4} \}$Those are...

View Article


Absolute convergence of series...

How do I prove that the following series converges absolutely$$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}e^{-(2n+1)a}e^{(2n+1)m(b-a)}$$ if we know that $b<0<a$.The first thing I tried is proving that...

View Article

Does $\lim_{z \to z_0} f(z) < \infty$ imply that $f(z)$ is bounded near...

Does $\lim_{z \to z_0} f(z) < \infty$ imply that $f(z)$ is bounded near $z_0$? For example, do we have that since$$\lim_{z \to 0} \frac{z}{\sin z} = 1$$we have $\frac{z}{\sin z}$ is bounded near $z...

View Article

Does this limit exist, how to prove it exists and what is its value? [duplicate]

$$\lim_{n\to\infty}\frac1n\sum_{k=1}^n\frac{\sin(\frac{\pi k^2}{2n})}{\sin\frac{\pi k}{2n}}$$

View Article
Browsing all 9124 articles
Browse latest View live


<script src="https://jsc.adskeeper.com/r/s/rssing.com.1596347.js" async> </script>