Articles of analysis

limit involving $e$, ending up without $e$.

Compute the limit $$ \lim_{n \rightarrow \infty} \sqrt n \cdot \left[\left(1+\dfrac 1 {n+1}\right)^{n+1}-\left(1+\dfrac 1 {n}\right)^{n}\right]$$ we have a bit complicated solution using Mean value theorem. Looking for others

Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$

Let $(X, Ω, μ)$ be a finite measure space. Assume that for any $t > 0$ there exists $E ∈ Ω$ satisfying $0 < μ(E) < t.$ Prove that for any $1 < p < ∞$ there exists a function $f ∈ L_p(μ)$ such that $f \notin L_q(μ)$ for any $q > p.$ I am […]

p-seminorms on smooth functions are equivalent

Let $K \subset R^n$ be compact, and let $C_0^{\infty}(K)$ be the space of smooth functions on with support in $K$. For $p \in [1,\infty), \alpha$ multiindex, let $|f|_{\alpha,p} = (\int_K |D^{\alpha}f|^p)^{\frac{1}{p}}$, and $|f|_{\alpha,\infty} = sup_K |D^{\alpha}f|$ – as usual. For $p \in [1,\infty]$, let $\tau_p$ be the locally convex topology generated on $C_0^{\infty}(K)$ by the […]

Limit of the sequence $\lim_{n\rightarrow \infty}n\left ( 1-\sqrt{1-\frac{5}{n}} \right )$, strange result

$\lim_{n\rightarrow \infty}n\left ( 1-\sqrt{1-\frac{5}{n}} \right )$ $\lim_{n\rightarrow \infty} n *\lim_{n\rightarrow \infty}\left ( 1-\sqrt{1-\frac{5}{n}} \right ) = \infty * \left ( 1-\sqrt{1-0} \right ) = \infty * 0 = 0$ Did I do it correctly? Problem is when I use my calculator and put big values for $n$, I get $2,5$ as result (if I take […]

A smooth function instead of a piecewise function

I want to find a smooth function approximating f(x) as best as possible: \begin{equation*} f(x) = \begin{cases} x & \text{if } x \le a,\\ a & \text{if } x > a. \end{cases} \end{equation*} as a smooth function ($a$ is a positive constants and x is a positive real number). $f(x)=\sqrt[n]{x}$ has a similar trend, but […]

The Dirac Delta Distribution $\delta_0 : D \to \mathbb R$ is not regular

I have a question on the following proof, that $\delta_0$ is not a regular distribution. We define $\delta_0$ as the linear function on test function with $$ \delta_0(\varphi) = \varphi(0) $$ for every test function $\varphi$ (note it is not defined as some function with $\delta(0) = 1, \delta(x) = 0$ for $x \ne 1$, […]

All derivations are directional derivatives

This question already has an answer here: Leibniz rule and Derivations 1 answer

Can I research in complex analysis, PDE and differential geometry without exposure to mathematical physics?

I love mathematics, but physics is far away from my interest. I see that recent mathematics research is strongly connected to mathematical physics which is something doesn’t interest me! I love mathematical entities and structures, stuff like relativity and quantum mechanics are not even close to my area of interest, so I wonder if I […]

Backwards heat equation (stability analysis)

Problem Consider the backwards heat equation of the form $$ \left\{ \begin{aligned} u_{t} &= \lambda^2 u_{xx}, & x\in[0,L], \quad t\in[0,T]\\ u(0,t) &= u(L,t) = 0 \\ u(x,T) &= f(x), \end{aligned} \right.\tag{*}\label{*}$$ Establish whether solution is unique and analyze its stability. Attempt of proving uniqueness My attempt to prove uniqueness is provided in this post. Attempt […]

Measuring the set-theoretical complexity of sets/spaces encountered in general analysis

In analysis, it is common to encounter subsets of $\mathbb R$ (or even $\mathbb R^n$) which appear to be “well-behaved”, especially with regard to properties like being measurable, compactness, etc. It is core to descriptive set theory (DST) that one is able to impose a classification of such subsets by closure properties of varying degree. […]