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).

learn more… | top users | synonyms (1)

1
vote
3answers
31 views

Is it possible to replace $\lim$ by $\sup$ here?

Let $(a_m)$ be a monotonically increasing sequence. Is it then possible that $\lim_{m\to\infty} a_m=\sup_m a_m$? I only know the fact that a bounded monotonically increasing sequence coverges to its ...
0
votes
1answer
20 views

Remember the implicit function theorem

First, I know the implicit function theorem, but unfortunately I always have to look it up again and again. If $F(x,y)=0$ then I always forget whether I have to invert the first matrix of the Jacobian ...
-2
votes
1answer
29 views

Prob. 3, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Can we find an example where $\mathbb{R}^3$ is a direct sum of two subspaces that are not orthogonal? A vector space $X$ is said to be a direct sum of two of subspaces $Y$ and $Z$ of $X$ if each $x ...
0
votes
1answer
26 views

compute very special limit in real number

Let the function $f:$ $\Bbb{R}\to\Bbb{R}$ such that $f(x)=\inf\{|x-me|:m\in\Bbb{Z}\}$ and consider sequence $\{f(n)\}$ then which of the following options is true? a) $\{f(n)\}$ is convergence b)the ...
-3
votes
1answer
27 views

Prob. 2, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $M$ be the subset of $\mathbb{C}^n$ such that $M$ consists of all $n$-tuples of $y = (\eta_1, \ldots, \eta_n)$ of complex numbers such that $\sum_{i=1}^n \eta_i = 1$. Then we can show that $M$ is ...
0
votes
1answer
22 views

Prob. 1, Sec. 3.3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Let $H$ be a Hilbert space, $M \subset X$ a convex subset, and $(x_n)$ a sequence in $M$ such that $\Vert x_n \Vert \to d$, where $d = \inf_{x \in M} \Vert x \Vert$. How to show that $(x_n)$ converges ...
0
votes
1answer
23 views

Compactness and convergence

Let $U$ be a subset of $\mathbb{R}^n$, and suppose that $U$ is not bounded. Construct a sequence of points $\{a_1, a_2, \ldots \}$ such that no subsequence converges to a point in $U$, then prove this ...
2
votes
1answer
29 views

Find all holomorphic functions, $f: \mathbb{C} \rightarrow \mathbb{C}$. so that $f'(0)=1$ and $f(x+iy)=e^{x}f(iy)$

Find all holomorphic functions, $f: \mathbb{C} \rightarrow \mathbb{C}$. so that $f'(0)=1$ and $f(x+iy)=e^{x}f(iy)$ I've been messing with this problem for most of today and haven't managed to get ...
0
votes
0answers
16 views

Fourier transform and series

Let $f \in L^2(\mathbb{R})$ and $F(f|_{[m,m+1]})$ be the Fourier transform of a restriction of $f$. Does this imply that $$\sum_{m,n \in \mathbb{Z}} |F(f|_{[m,m+1]})(2 \pi n)|^2 $$ exists and is ...
0
votes
0answers
7 views

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then$ A\in J \implies f(A)$

Can anyone help me prove: If $U$ is an open set in $R^n$, $f:U->R^n, f\in C^1$ , $A\subset R^n, det(f')_{|intA} \neq 0.$ Then $ A\in J \implies f(A)\in J$; $J$- set are Jordan measurable sets in ...
2
votes
0answers
71 views

Are “Transition Books” (Spivak/Apostol/Courant) really necessary?

Why do so many people recommend Spivak, Apostol, and Courant calculus textbooks, especially as a preparation toward the advanced courses like analysis and abstract algebra? Are they really necessary? ...
2
votes
2answers
91 views

Derivative of the power tower

May somebody help me to correctly calculate the dervative of the $n$-th power tower function? $$ \begin{align} f_1(x)&=x\\ f_n(x)&=x^{f_{n-1}(x)}\\ &=x^{x^{x^{...^x}}}\text{ where ...
0
votes
0answers
14 views

Product of product-measurable function and measurable function product-measurable?

Given two measurable spaces $(\Omega, \mathcal{F}), (\Theta, \mathcal{F}_\Theta)$ and their product with the product-sigma-algebra $(\Omega \times \Theta, \mathcal{F} \otimes \mathcal{F}_\Theta)$ and ...
0
votes
0answers
10 views

When do closed subspaces of a Banach space fit together nicely?

Let $E$ be a Banach space, and let $F_1, F_2, F_3, ...$ be a sequence of closed subspaces of $E$ with $F_i\cap F_j=0$ whenever $i\neq j$. Denote by $\sum_n F_n$ the (not necessarily closed) subspace ...
3
votes
2answers
40 views

Upper and Lower Darboux integral of a piecewise function $f(x)=x$ and $f(x)=0$.

Let $0<a<b$. Find the upper and lower Darboux integrals for the function $$f(x)=x$$ if $x\in[a,b]\cap\mathbb{Q}$ and $$f(x)=0$$ if $x\in[a,b]-\mathbb{Q}$. I am so lost on this problem. Any ...
-1
votes
0answers
25 views

Inequality with poisson r.v. [on hold]

Let $r>0$ and $X \sim Poisson(\lambda)$. Prove that ( $e=2.71...$) $$ \mathbb{E} X^r \le r^r + (e \cdot \lambda)^r $$ I can show it for $r \in \mathbb{N}$ by writing expected value as series, ...
-2
votes
0answers
66 views

Is it possible to find $\lim_{n\rightarrow\infty}\frac{1^n+2^n+3^n+\ldots+n^n}{n^{n+1}} $ without using integral or combinatory logic? [on hold]

$$\lim_{n\rightarrow\infty}\frac{1^n+2^n+3^n+\ldots+n^n}{n^{n+1}} $$ I apologize for asking the same question again, but I wanted to ask something. Is there a possibility that this problem could be ...
0
votes
0answers
22 views

Show uniform continuity $\sum_{n=1}^{\infty}{\cos(nx)(nx)^{1\over 2}\over{(n+1)^2}}.$

Show that the series is uniformly convergent in any interval $[0,a]$ for $a>0$. $$\sum_{n=1}^{\infty}{\cos(nx)(nx)^{1\over 2}\over{(n+1)^2}}.$$ I'm not sure what to do. I think the Weirstrass M ...
0
votes
1answer
22 views

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $-\Delta u=1$ in $\mathbb{R}^n$?

Is there any nonnegative $u\in C^2(\mathbb{R}^n)$ with $\Delta u=-1$ in $\mathbb{R}^n$? I think not, but how can we prove it? Let's assume that such a solution exists. Let $R>0$ and $B_R:=B_R(0)$ ...
2
votes
1answer
37 views

The Lebesgue-Borel measuref the difference between two open balls tends to $0$ as the radii tend to $\infty$

Let $\lambda_n$ be the Lebesgue-Borel measure on the Borel-$\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ and $x,y\in\mathbb{R}^n$. What is the easiest way to prove $$\frac ...
0
votes
2answers
50 views

function which is Riemann integrable

Consider $f:[-1,1]\to\mathbb{R}$, $x\mapsto \begin{cases} 1, & \text{if } x=0 \\ 0 & \text{else } \end{cases}$ I want to know why f is Riemann integrable and I tried something. First of ...
2
votes
2answers
65 views

Find the limit of $\lim_{n\to \infty}n^2({1\over{n^3+1^3}}+{1\over{n^3+2^3}}+\cdots+{1\over{n^3+n^3}}).$

Find the limit of $$\lim_{n\to \infty}n^2({1\over{n^3+1^3}}+{1\over{n^3+2^3}}+\cdots+{1\over{n^3+n^3}}).$$ I'm not sure how to evaluate this limit. Any hints or solutions are greatly appreciated. I ...
1
vote
0answers
20 views

Difficult examples of invertible, differentiable functions

Give an example of: 1)$f:\mathbb R^2 \to \mathbb R^2$ such that $f$ is invertible in some neighbourhood of $x_0$ (that is $f$ is locally invertible) but $|Jf(x)|=0$ (jacobian determinant) $\forall x$ ...
2
votes
1answer
24 views

Example of continuous curve $f:[0,1]\to\mathbb{C}$ for which $f(0)=0,f(1)=1$ which has no point which satisfy certain conditions?

Does there exist any continuous curve $f:[0,1]\to\mathbb{C}$ for which $f(0)=0,f(1)=1$ and for which there is no pair of points $p,q\in f([0,1])$ such that $q-p=0.75$?
7
votes
3answers
132 views

What are the differences in mental skills required to master abstract algebra and analysis?? [on hold]

I had took undergraduate-level abstract algebra and analysis courses before. And I find I can do proofs in analysis faster than in abstract algebra. However some other students is opposite to me. I ...
2
votes
2answers
57 views

Show that $\ln(1+x)\leq x-{1\over 2}x^2+{1\over 3}x^3$.

Prove that for $x\in (0,\infty)$, $\ln(1+x)\leq x-{1\over 2}x^2+{1\over 3}x^3$. I'm a little bit stuck, but I think I have the right idea. Any hints or solutions are greatly appreciated. Here is what ...
0
votes
1answer
37 views

Expansion of $(1+x)^\alpha$.

Let $\alpha\in\mathbb{R}$. Prove that, for all $x \in [0, 1)$ we have $(1 + x)^\alpha=1+{\alpha x\over 1!}+{\alpha(\alpha-1)x^2\over 2!}+\cdots+{\alpha(\alpha-1)\cdots(\alpha-k+1)x^k\over k!}+\cdots$. ...
2
votes
3answers
67 views

Limit $(0÷0)$ and use L'hopital Rule

find value of the $$\lim_{x\to0}\frac{e-(1+x)^{\frac{1}{x}}}{x}$$I use hospital law and can't find answer
0
votes
1answer
24 views

Manually plotting some particular graphs

How to plot graphs like these manually: 1) $f(x)=\ln(1+x^2)$ 2) $f(x)=\frac8{2+x^2}$ 3) $f(x)=\frac{\sin x}{\sqrt{1+\tan^{2}x}}+\frac{\cos x}{\sqrt{1+\cot^{2}x}}$ I have no idea how to plot the ...
0
votes
2answers
23 views

Norm on the space of sequences

Given the sequence spaces $\ell^p$ that are defined as: $$\ell^p = \left\{a = (a_n)_{n\in\mathbb{N}}, \sum_{n=0}^\infty |a_n|^p < \infty\right\}$$ for $\infty > p ≥ 1$, I'm trying to show that ...
1
vote
2answers
31 views

Solutions for the following equation.

Let us have an $n$ positive integer. How many solutions do we have for the following equation in the interval $(0,\frac{\pi}{2})$? $$\underbrace{\cos(\cos(\ldots(\cos x)\ldots))}_{n\text{ times ...
0
votes
1answer
25 views

Compute the wedge product n times

Let $\omega$ be a 2-differential form in $\mathbb{R}^{2n}$ given by $$\displaystyle \omega=dx^1\wedge dx^2+dx^3\wedge dx^4 + \cdots + dx^{2n-1}\wedge dx^{2n}$$ Compute: $$\displaystyle ...
0
votes
1answer
28 views

How to show that $\dfrac{\sin(x^2+y^2)}{(x^2 + y^2)^\alpha}$ integrable on $\mathbb{R}^2$

I need to show that $$k(x,y) = \dfrac{\sin(x^2+y^2)}{(x^2 + y^2)^\alpha}$$ is integrable on $\mathbb{R}^2$ for $1<\alpha <2. $ How do I go about this? I'm pretty sure I need to use Tonelli's ...
0
votes
1answer
36 views

Prove limit exists if and only if left and right limits exist and are equal

Prove $\lim_{x\to a}f(x)=L \iff \lim_{x\to a^+}f(x)=L=\lim_{x\to a^-}f(x)$ I have no problem with the $(\Leftarrow)$ direction but I can't do it for the other direction. Proofs for both directions ...
0
votes
1answer
18 views

Holder's inequality. Proof using conditional extremums .Need help, can't see how one step is found.

Prove:$$\sum_{i=1}^{n}a_ix_i\leq (\sum_{i=1}^{n}a_i^p)^{1\over p}(\sum_{i=1}^{n}x_i^q)^{1\over q} $$ $(a_i\geq0,x_i\geq0,i=1,..n,p>1, {1 \over p}+{1\over q}=1)$ Let ...
2
votes
1answer
57 views

All derivations are directional derivatives [duplicate]

Let $X : C^{\infty}(\mathbb{R}^n) \rightarrow \mathbb{R}$ be a derivation, so i.e. linear and satisfying the Leibniz Rule $$X(fg)=X(f) \cdot g(a)+X(g) \cdot f(a)$$ for some fixed $a \in ...
0
votes
0answers
47 views

On the definition of limit

Let $f:A\subset\mathbb R\to\mathbb R$ be a function and $a\in\mathbb R$. I want to know which one of the following is the definition of $\lim_{x\to a} f(x)=L$: $\forall\epsilon> ...
0
votes
0answers
20 views

A continuous injection

Can i prove that $W^{1,p}_0\hookrightarrow L^q_{\frac{N}{p}}$ continuously where $\Omega\subset \mathbb{R}^N$ is smooth and bounded and $L^2_{\frac{N}{p}}$ is equiped with the norme ...
1
vote
1answer
11 views

Describing convergence/divergence of a complex sequence

Let (a$_n$)$_{n \in N}$ be a complex sequence and a $\in$ C. Show that the following statements are equivalent: $\forall$ $\varepsilon$ > 0 $\exists$ N $\in$ N $\forall$ n $\geq$ N : |a$_n$ - a| ...
0
votes
1answer
16 views

sequence spaces as subsets of each other

Given the sequence spaces $\ell^p$ that are defined as: $$\ell^p = \left\{a = (a_n)_{n\in\mathbb{N}}, \sum_{n=0}^\infty |a_n|^p < \infty\right\}$$ for $\infty > p ≥ 1$, how can it be shown ...
-1
votes
1answer
13 views

The truncation error associated with linear interpolation of a function $f(x)$ using ordinates $x_0$ and $x_1$ is not larger in magnitude [on hold]

To show that the truncation error associated with linear interpolation of a function $f(x)$ using ordinates $x_0$ and $x_1$ is not larger in magnitude than $$\frac {(x_1 - x_0)^2}{8} \times \max_{x ...
0
votes
0answers
7 views

If $S_M$ denotes the measure of a submanifold, then $\frac 1{r^{n-1}n\omega_n}\int_{\partial B_r}u(x)\;dS_{\partial B_r}(x)\to u(y)$ for $r\to 0$

Let $S_M$ denote the "surface measure" of a submanifold $m$ $B_\varepsilon(y)$ denote the open ball around $y$ with radius $\varepsilon>0$ $\omega_n$ denote the volume of the $n$-dimensional unit ...
0
votes
0answers
19 views

What does it mean for a sequence of complex functions to converge uniformly? [on hold]

Let E $\subset$ C. What does it mean for a sequence of complex functions with domain E to converge uniformly? Give an example of a sequence of functions with domain C that converges but not uniformly. ...
1
vote
0answers
44 views

Compute the double Hodge star operator

I am taking a course in Multivariable Analysis and I am asked to do the following problem: Show that $\ast\ast\omega = (-1)^{k(n-k)}\omega$ So I start as follows: We know that $\displaystyle ...
3
votes
0answers
36 views

Prob. 10, Sec. 3.2 in Erwine Kreyszig's “Introductory functional analysis with applications”

Here is Prob. 10 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: ... Let $T \colon X \to X$ be a bounded linear operator on a complex ...
1
vote
0answers
22 views

Prob. 9, Sec. 3.2 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS

Here is Prob. 9 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $V$ be the vector space of all continuous complex-valued functions on ...
0
votes
1answer
25 views

limit involving a Taylor Polynom

Let $I \subset \mathbb{R}$ be an interval, and let $f: I \to \mathbb{R}$ be a function that's at least n-times differentiable. It needs to be shown that if a polynomial $P(x)$ is of degree $≤ n$, and ...
4
votes
4answers
61 views

$x \perp y$ if and only if $\Vert x + \alpha y \Vert \ge \Vert x \Vert$ for all scalars $\alpha$

Here's Prob. 8 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Show that in an inner product space, $x \perp y$ if and only if $\Vert x + ...
0
votes
1answer
35 views

find special limit in infinite and compute

find $$\lim_{n\to\infty}\left( \frac{a^\frac{1}{n}}{n+1}+ \frac{a^\frac{2}{n}}{n+\frac{1}{2}}+\cdots +\frac{a^\frac{n}{n}}{n+\frac{1}{n}}\right)$$ when $n\to\infty$ please help me to find it.when ...
1
vote
1answer
20 views

Prob. 6, Sec. 21 in Munkres' TOPOLOGY, 2nd ed: How to show directly that this sequence of functions does not converge uniformly?

For each $n = 1, 2, 3, \ldots$, let $f_n \colon [0,1] \to \mathbb{R}$ be defined by $$f_n(x) \colon= x^n \ \ \ \mbox{ for all } \ x \in [0,1].$$ Then $$ \lim_{n \to \infty} f_n(x) = \begin{cases} ...