Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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$f$ is continuous on $E$ if and only if its graph is compact.

This question may be asked before under different formulation, the original problem is Chapter 4, Exercise 7 of Rudin's text: The Principles of Mathematical Analysis: Problem: If $f$ is defined on ...
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2answers
12 views

Proving Lower bounds on an Approximately Linear Function

We are looking for a lower bound on the function, $\frac{1.31}{e^{\frac{1.31}{x+1}} - 1}$ for $x \geq 2$. This function seems to behave linearly. We believe that the following statement holds: ...
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19 views

How to prove tha is a self-adjoint operator?

I have this operator from $H^1_0$ to $H^1_0$ defined by: $Au(t)=\int_0^1 G(t,s) f(s,u(s))ds$ where $G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$ i want to see if ...
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15 views

Closed form defines locally a function?

In my particular example, I have a volume form on a one-dimensional manifold, so i.e. this volume form $dx$ is closed. Now, I was wondering, does the integral function $f(y):=\int_{x_0}^{y} dx$ define ...
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14 views

Covariant and partial derivative commute?

I know that we have for a function $\Gamma: (-\varepsilon,\varepsilon)^2 \rightarrow M$ $$\nabla_{\frac{\partial \Gamma(s,t)}{\partial s}} \frac{\partial \Gamma(s,t)}{\partial t} = ...
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19 views

Jacobi fields and variations

I want to show that every Jacobi field is a variation of geodesics, i.e. let $Y: I \rightarrow TM$ be a Jacobi field along a geodesic $\gamma$, then I want to show that $Y$ can be written as a ...
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1answer
45 views

Every neighborhood $N_r(x)$ in $\mathbb{R}^n$ is connected

I am working on an exercise in baby Rudin (Ex 2.20 in particular) and as part of that I am trying to show that any neighborhood in a metric space is connected. I've seen several differing definitions ...
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If I have let's say a finite number of points in a metric space, and lets say that space has a couple of different metrics.

If I have let's say a finite number of points in a metric space, and lets say that space has a couple of different metrics defined. Is the diameter of a subset unique with respect to the two most ...
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1answer
24 views

Why do we study multi-valued(set valued) mappings?

I am working on a problem that has to do with multi-valued mappings. Precisely, Iterative methods for Fixed points for multivalued mappings. However, I have no clear motivation for studying such ...
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20 views

If $\partial E$ has Jordan outer measure zero, then $E$ is measurable.

I am going through Tao's measure theory book, and have to prove If $\partial E$ has Jordan outer measure zero, then $E$ is measurable. where $\partial E$ denotes the boundary of the set $E$. I ...
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0answers
28 views

Convergence of a sequence of integration

I am considering one problem and I am stuck in this step. The problem is that What conditions on function $f(u,\epsilon)$ are required to satisfy $$ \int_0^\epsilon f(u,\epsilon)\,du \rightarrow 0 ...
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0answers
40 views

Inner Product Properties And Applications

In every calculus or analysis class we are told that the concept of inner product is very important, and that its applications are vast, diverse, and extremely useful. I don't think there is a single ...
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6 views

About a property of the upper triangular projection of a matrix

I need a hand checking that a property about the upper triangle projection of an infinite matrix holds. $\bullet$ Let A be an infinite matrix $A=(a_{ij})_{i\geq 1\;j\geq 1}$. We define its upper ...
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1answer
14 views

$1$-form of a antiholomorphic function, Cauchy-Goursat Theorem

Let be $f:U\to \Bbb C$ antiholomorphic function. Show that the 1-form $f(z)d\overline{z}$ is closed. We have that $\overline{f}$ is a holomorphic function, so by Cauchy-Goursat Theorem the ...
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1answer
44 views

Schwarz Lemma of Complex Analysis

Let be $f:B(0,1)\to B(0,1)$ holomorphic function such that $$f(0)=f'(0)=\cdots=f^{(n-1)}(0)=0$$ but $f^{(n)}(0)\neq 0.$ Show that $|f(z)|\le |z|^n,$ for every $z\in B(0,1)$ and $|f^{(n)}(0)|\le ...
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12 views

Inverse Laplace operator $\Delta^{-1}$ in $H^1_0, \ H^{−1}$.

Let open domain $\Omega \subset \mathbb R^n$, $u : \Omega \rightarrow \mathbb R$, $u \in H^1_0 (\Omega)$ and $f \in H^{−1}(\Omega)$. I'm looking for some known expressions of the inverse Laplace ...
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3answers
78 views

Show that $\frac{n}{n^2-3}$ converges

Hi I need help with this epsilon delta proof. The subtraction in the denominator as well as being left with $n$ in multiple places is causing problems.
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0answers
22 views

Operator Norm and Submultiplicativity against the Spectral Norm

Consider $\mathcal{A}:\mathbb{R}^{n\times m}\to \mathbb{R}^{p\times q}$ to be a linear operator. I know that by considering the trace norm and using the submultiplicativity of the operator norm we ...
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35 views

local inverse functions

consider $f(x,y)=(x\sin y,x\cos y),\; (x,y)\in (0,\infty)\times (0,3\pi)=U$. f is locally invertible at every point in U, because $\det(Df(x,y))\not= 0$ for all $(x,y)\in U$. I want to know : What are ...
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0answers
33 views

Jacobi field strange condition.

I am currently reading a textbook (Kuehnel) saying that if $V,W \in T_pM$ are such that $\langle V,W \rangle =0$ and $\|V\|=\|W\|=1,$ then $Y(t):=D \exp(tV)(tW)$ is a Jacobi field. The thing is, I ...
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5answers
125 views

Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$”

If $C$ and $A$ are the circumference and area of a circle, then $$ \frac{C^2}{A}=4 \pi\; . $$ I'm looking for a reference to an elementary but rigorous proof of it. I'm particularly interested in ...
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1answer
46 views

Assumptions on functions so that integral is zero

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two arbitrary functions. Assume $g\in L^2(\mathbb{R})$. I'm looking to find out the minimal set of assumptions on $f$ and $g$ such ...
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1answer
41 views

convolution and integral limits

Let $\xi$ be an increasing function , and $f$ be a continuous function on the interval $[0,1]$. Take $\phi$ a smooth function such that $\int_0^1 \phi(s)\, ds= 1 $ and consider an approximation of ...
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1answer
27 views

Local Lipschitz continuity

In some proof I have seen the author use that if $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and bounded, then it is locally Lipschitz continuous. I have never seen that before and I don't find ...
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64 views

Show a function defined by summation is increasing, another is decreasing

Problem: For real numbers $x\ge1$ and $k>0$, let $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as follows. $f(x) = -\frac{1}{x}+\sum_{n=1}^{\infty}\frac{1}{(nk+x)^2}$ , ...
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1answer
57 views

Zeros of an analytic function [duplicate]

How to prove zeros of a real analytic function (non-zero function) is always countable?
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1answer
61 views

Rigorously proving $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx= \frac{\pi}{2}$ [duplicate]

I want to prove the famous formula: $\displaystyle\int^{\infty}_0 \frac{\sin{x}}{x} dx = \frac{\pi}{2}.$ There are many ways to do it, for example, by some Fourier analysis. But how about a simple ...
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1answer
23 views

Show that the Lebesgue Stieltjes measure corresponding to $\alpha(x) = \mu((0,x])$ is $\mu$.

This is exercise 4.1 from Bass: Let $\mu$ be a measure on the Borel $\sigma$-algebra fo $R$ such that $\mu(K) < \infty$ whenever $K$ is compact, define $\alpha(x) = \mu((0,x])$ if $x \ge 0$ and ...
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1answer
40 views

Is $C^{\infty}$ dense in $W^{k,p}$?

The $C_c^{\infty}$ are certainly not sense in an arbitrary $W^{k,p}$ space. Despite, I started wondering whether at least $C^{\infty}$ is dense? Now, this can certainly not be true for general ...
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0answers
9 views

Calculate factor for FWHM in a sech(x)-function

I have a $sech\left(\frac{\pi}{2}a\cdot x\right)$-function, and I want to calculate $a$ such that the FWHM of the function meets a specific width $\Delta x$. So I started with ...
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39 views

Research areas lying at the confluence of Analysis and Geometry [on hold]

I wanted to get expert opinion on what are the areas of active research lying at the confluence of Analysis and Geometry. Two areas that I have heard about are : (1)Geometric Analysis and ...
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1answer
26 views

ODE, Picard approximation of a second order equation: How do I make sure that this is correct.

I have the following problem: $$\ddot{x} + \dot{x}^2-2x=0$$ and I.V are: $x(0)=1 \qquad$ $\dot{x}(0) = 0$. and I need to find two first "Picard" approximations. I first arranged it in the form ...
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1answer
67 views

Is the space of continuous functions with bounded variation separable?

Let $$BVC([0,1]) = \{f:\mathbb [0,1] \to \mathbb R,\, f\in BV([0,1])\cap C([0,1]),\, f(0)=0\},$$ and $$\|f\|_{BVC}=\mathrm{Var}(f).$$ Is $BVC([0,1])$ separable?
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1answer
24 views

Nest of intervals explanation

I am currently reading Konrad Knopp book about infinite series, I just don't get the part where he mentions that the nest of intervals would determine or define as he said a rational number s if it ...
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24 views

Methods of Real Analysis solution book [on hold]

Where can I find a solutions book for Methods of Real Analysis by RIchard Goldberg?
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2answers
38 views

When is a continuous function piecewise monotone?

Given a continuous function $f:[a,b]\mapsto \mathbb{R}$, are there known additional conditions that ensure $f$ is piecewise monotone? Like this question, my motivation is to decompose the interval ...
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1answer
14 views

For every $n \ge 2 $ , existence of uncountably many mutually disjoint closed balls , complement of whose union is path connected

For every $n \ge 2 $ , does there exist an uncountable family $\{D_\lambda: \lambda \in \Gamma\}$ of mutually disjoint closed balls in $\mathbb R^n$ , such that $\mathbb R^n \setminus \cup_{\lambda ...
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2answers
37 views

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ?

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ? Or , does every infinite dimensional normed linear space has ...
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0answers
24 views

The question about the support of Fourier transform of $|f|^p$

Suppose $f$ is a smooth function with $\mathbb{supp}{(\mathcal{F}{f})} \subset B(0,1)$. In addition, assume $f$ is non-negative. We can observe that the function $|f|^2$ has a nice property : ...
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0answers
25 views

How to find the domain of the support function

First the definition of the support function of set $S$: Let $S$ be a nonempty convex set. The support function $h$ of $S$ is the real-valued function defined by $$h(x)=\sup_{s\in S}\langle ...
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1answer
31 views

Show that $\|f\|^{2}$ attains a minimum value on the interior of $B$

I am looking for any help, hints, or suggestions in how to go about this problem from a previous qualifying exam. We are given a smooth mapping $f: U \rightarrow \mathbb{R}^{n}$ whose differential ...
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3answers
63 views

History of convolution

Let $f, g\in L^{1}(\mathbb R),$ we may define the convolution of $f$ and $g$ as follows: $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy, (x\in \mathbb R).$ It is well known that it can be defined on ...
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1answer
42 views

What is the meaning of $\lim_{\Delta(P) \to 0} F(P) = L$ for partitions

Let $[a,b]$ be an interval, and denote by $\mathcal P[a,b]$ the family of all partitions of $[a,b]$, i.e. sets $P = \{ a = x_0 < x_1 < \ldots < x_n = b \}$. For some $P \in \mathcal P[a,b]$ ...
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1answer
31 views

Is the integral with respect to increasing continuous functions the limit of integrals with respect to $C^1$ functions?

if $\xi$ is continuous increasing can we find $\xi^n\in C^1$ such that $$\int_0^t f(u)\, d\xi = \lim_n\int_0^t f(u)\, d\xi^n$$ for every continuous $f$?
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1answer
29 views

Unit radial vector field

Lee's book defines the unit radial vector field in normal coordinates as $$ \partial_r:= \frac{x^i}{r(x)} \partial_i$$ and $r(x):=\sqrt{\sum_i (x^i)^2}$ Now this is a unit vector field iff ...
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1answer
24 views

Complement of the union of finitely many , mutually disjoint , closed balls in $\mathbb R^n$ , for $n>1$ , path connected? [on hold]

Let $D_1,D_2,...,D_k$ be finitely many , mutually disjoint , closed balls in $\mathbb R^n$ , where $ n \ge 2$ . Then is $\mathbb R^n \setminus \cup_{i=1}^k D_i$ path connected ?
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set of all accumulation points of A is countable

How do I approach this question stating: Construct a compact set A of R such that the set of all accumulation points of A is countable. F compact means closed and bounded. Let $x_k$ element in it. ...
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1answer
39 views

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma

Prove that $\{ar+b:a,b\in Z\}$, where $r$ is an irrational number is dense in $R$ by using the following lemma: If $x$ is irrational, there are infinitely many rational numbers $h/k$ with $k\gt 0$ ...
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48 views

Is this sequence is dense?

Define $S _m, _n = $ n th smallest square number which is bigger or same than $10^ {m-1}$and smaller than $10^m$ Then is the sequence $ \frac{S_m,_n} {10^m}$ is dense in (0,1) or arbitary ...
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3answers
40 views

Convergence of improper integral?

Consider an improper integral such that: $$I = \int_0^{+\infty} \frac{f(x)}{x}dx.$$ If $\int_0^{+\infty}f(x)dx < + \infty$, Can we conclude that the integral I converges? Thanks for any answer or ...