Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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6
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5answers
2k views

Need Help: Any good textbook in undergrad multi-variable analysis/calculus?

This semester, I will be taking a senior undergrad course in advanced calculus "real analysis of several variables", and we will be covering topics like: -Differentiability. -Open mapping theorem. ...
9
votes
0answers
529 views

Cauchy-Formula for Repeated Lebesgue-Integration

Recently, I came across the following statements. They were annotated as consequences of Fubini's Theorem but neither proof nor reference were given. Let $f:[a,b]\times [a,b]\to\mathbb{R}$ be ...
7
votes
3answers
447 views

Cauchy Sequence in $X$ on $[0,1]$ with norm $\int_{0}^{1} |x(t)|dt$

In Luenberger's Optimization book pg. 34 an example says "Let $X$ be the space of continuous functions on $[0,1]$ with norm defined as $\|x\| = \int_{0}^{1} |x(t)|dt$". In order to prove $X$ is ...
2
votes
3answers
263 views

How to prove $(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$ for $x \in [0,1)$?

I tried to prove that $$(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$$ for $x \in [0,1)$ with Fourier analysis, but I just found a Fourier series which defines the function. I also found the ...
12
votes
1answer
693 views

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic

If $1\leq p < \infty$ then show that $L^p([0,1])$ and $\ell_p$ are not topologically isomorphic Maybe I would have to use the Rademachers.
4
votes
1answer
522 views

An existence of global solution of differential equation of first order

Let $f: (a,b) \times \mathbb{R} \rightarrow \mathbb{R}$ be of class $C^1$ in $D:=(a,b) \times \mathbb{R}$ and satisfies condition $$| f(t,x)| \leq A+B|x| \textrm{ for } (t,x) \in D,$$ where $A,B$ ...
1
vote
1answer
181 views

Problem with Picard Iteration

I have $ \frac{dy}{dx} = y^2, y(0) = y_0 $ I have solved this as $y = \frac{y_0}{1 - x y_0}$ Which has the Taylor expansion $ y_0+y_0^2 x+y_0^3 x^2+y_0^4 x^3+y_0^5 x^4+ ...$ However, when I ...
2
votes
2answers
283 views

Convergence of the series $\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$

Could you help me to understand for which $x$ this series converge $\displaystyle\sum \frac{\sqrt{n+1}-\sqrt{n}}{n^x}$?
2
votes
2answers
183 views

$f(x,y)$ in polar coordinates

So, I have $ f(x,y) = (x^2-y^2, 2xy) $, which is a local $\mathcal C^1$ isomorphism in $\mathbb R^2 \setminus \{(0,0)\}$. I have to write this function in polar coordinates: $$f(x,y) = ...
7
votes
3answers
553 views

Do continuous linear functions between Banach spaces extend?

Just wondering... Let $E$, $G$ be Banach spaces, let $U\subset E$ be a subset of $E$, and let $f:U\rightarrow G$ be a continuous linear function. Can $f$ be extended to a continuous linear function on ...
2
votes
1answer
203 views

What is this series called and when does it diverge?

What is this series called (if it has a name)? When does it diverge without analytic continuation and when does it diverge with analytic continuation? $\sum_{k_1,\dots,k_m=1}^{\infty} ...
0
votes
1answer
472 views

Could explain me how eigenvector helps with compute gradient and how make differentiate operation on decrete space like digital image?

Could you explain me how eigenvector helps with compute gradient and how make differentiate operation on descrete space like digital image? I know that this question is so connected with computer ...
2
votes
0answers
181 views

Solution to polynomial equations with non-radicals

For degree 1, 2, 3 and 4 there is an "extended a,b,c-formula" (like the one we learn in middle or high school, http://en.wikipedia.org/wiki/Quadratic_equation) for the solution to a polynomial ...
0
votes
1answer
73 views

finding the function

Define a function $f\colon\mathbb{R}\to\mathbb{R}$ which is continuous and satisfies $$f(xy)=f(x)f(y)-f(x+y)+1$$ for all $x,y\in\mathbb Q$. With a supp condition $f(1)=2$. (I didn't notice that.) How ...
2
votes
1answer
182 views

Requirement of closed and bounded set $[a,b]$ in the Ascoli theorem

In Wikipedia, the Ascoli theorem requires the functions to be continuous on the closed and bounded interval $[a,b]$. However, in the proof given in the book "Theory of Ordinary Differential ...
4
votes
0answers
158 views

Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion

Let $ 0 < r < 1$, fix $x > 1$ and consider the integral $$ I_{r}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$ In the investigation of ...
5
votes
1answer
186 views

Can the Fourier transform be defined as an integration over $\mathbb C$ instead of $\mathbb R$?

Can the Fourier transform of a whole function $f:\mathbb R\mapsto\mathbb C$ be defined as integration over $\mathbb C$ instead of $\mathbb R$ as well, such that $$\tilde f(k) = \frac{\mathcal ...
1
vote
1answer
67 views

weak solutions need to have local integrability condition?

I am currently studying Poisson and Laplace equations. This is just a small question that has been causing me some confusion, and I would like some clarification before I resume my study. For example, ...
0
votes
2answers
518 views

The primitive of a discontinuous function?

I'm thinking about the primitive of an arbitrary function. As we all know, every continuous function has a primitive according to the Newton-Leibniz formula, $\int_{x_0}^{x} f(x) dx$ is the primitive ...
1
vote
0answers
97 views

Simple Swapping Integral Question (A Proof in Univalent Functions by Pommerenke)

This is really a simple question involving swapping the order of integration. The step I'm confused about comes from a proof in Pommerenke's "Univalent Functions," and for those of you with a ...
3
votes
4answers
1k views

Every Cauchy sequence in a metric space is bounded

Is the following correct or along the right lines? Thanks for any help Question A sequence $\{x_n\}$ in a metric space is said to be bounded if it is contained in some open ball $B(a,r)$. Prove ...
-1
votes
3answers
206 views

Why don't we define the class of $C^{\infty}$ in this way?

$C^{\infty}$ is defined to be the class of functions which have all orders of derivative. But as a convention, as far as the infinity is concerned, we always refer to limit. So why don't consider the ...
2
votes
2answers
168 views

Does anyone know how to prove this inequality

Does anyone know how to prove the following inequality ...
1
vote
0answers
385 views

Taylor theorem for function of several variables

Where one could find the proof of the following version of Taylor theorem for functions of several variables? Assume that $f$ is a function of class $C^{n+k}$ defined in a neighbourhood $W$ of zero ...
13
votes
2answers
239 views

Existence of some sort of 'infinite algebraicity' of transcendental numbers

Given an arbitrary number, say, $\alpha \in \mathbb{C}$, can anyone supply either (a) a reason for the existence (or general non-existence) of, or (b) the reverse engineering of a (convergent) ...
0
votes
1answer
76 views

Rate of convergence of double sequences

Suppose $ \{X_{n,m} \}$ be a double sequence of real numbers and suppose $\lim_{n}\lim_{m}X_{n,m}=X$. What is the definition and reference for the rate of convergence of double sequences?
7
votes
4answers
922 views

How many smooth functions are non-analytic?

We know from example that not all smooth (infinitely differentiable) functions are analytic (equal to their Taylor expansion at all points). However, the examples on the linked page seem rather ...
6
votes
2answers
1k views

Differentiation under the integral sign for Lebesgue integrable derivative

The problem is the following: Let $a,b,c,d \in \mathbb R$ be given such that $a<b$ and $c<d$. Suppose $f: [a,b]\times [c,d] \to \mathbb R$ is a function such that $\partial_1 f: [a,b]\times ...
101
votes
5answers
7k views

What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
15
votes
2answers
557 views

Smoothing a Sobolev function

Let $u \in H^1({\mathbb R}^n)$, $n \geq 2$. Let $\varphi \in C^\infty_0({\mathbb R}^n)$ with $\varphi \geq 0$. Let $\eta$ be a smoothing kernel with $\eta \in C^\infty_0({\mathbb R}^n)$, $\eta \geq ...
5
votes
4answers
474 views

History of analysis?

Any sites detailing the history of analysis post 1820 (to mid 1900s?) - vis-à-vis Cauchy, Weierstrass, Riemann, Bolzano, ..., Kuratowski, Hilbert? It's something that appears quite interesting and I ...
1
vote
1answer
244 views

Theorem 191 from the book Inequalities of Hardy, Littlewood and Pólya.

In the pic, in the second proof of Thm 191 (the one that starts at the paragraph:"We can prove theorem 191 without appealing to the more difficult theorem 190..."), I don't understand why: $$\int ...
3
votes
3answers
339 views

Inequality for logarithms

I conjecture the following inequality is true $$\ln x \le (x - 1)\ln\frac{x}{x-1}$$ for all $x > 1$, but I cannot give a proof. I will appreciate if someone can provide one.
5
votes
1answer
103 views

How to show $\int_{\mathbb{R}^d} u(x) e^{-|x|^2} (e^{-|x|^2 / n} - 1)^2 dx \rightarrow 0$, if $u\in L^2(\Bbb R^n)$?

How can I prove that $$ \int_{\mathbb{R}^d} u(x) e^{-|x|^2} (e^{-|x|^2 / n} - 1)^2 dx \rightarrow 0 $$ as $n \rightarrow \infty$? Here $u \in L^2(\mathbb{R}^n)$. I'm thinking the dominated ...
2
votes
1answer
197 views

L'Hospital's Rule for $\infty/\infty$

Arthur Mattuck in his Introduction to Analysis book, pg. 220 says, in order to prove L'Hospital's Rule for $\infty/\infty$ case, Let $L=\lim_{x \to \infty} \frac{f'(x)}{g'(x)}$ and choose $a$ ...
7
votes
1answer
272 views

Locally Lipschitz implies Lipschitz under equivalent metrics?

Bonjour. For $i=1,2$ let $X_i$ be a non-empty set and $d_i$ a metric $X_i^2 \to \mathbb{R}$. Suppose $f$ is a locally Lipschitz (*) function $(X_1, d_1) \to (X_2, d_2)$. Question. Do there exist ...
4
votes
1answer
285 views

Evaluating a limit of the truncated exponential series motivated by the prime number theorem for $k$ distinct prime factors.

If $\pi_k(n)$ is the cardinality of numbers with k factors (repetitions included) less than or equal n, the generalized Prime Number Theorem is: $$\pi_k(n)\sim \frac{n}{\ln n} \frac{(\ln \ln ...
2
votes
1answer
405 views

Primes and Riemann zeta function.

Primes numbers and Riemann zeta function. Question 1: Is there a proof of the infinitude of prime numbers using the Riemann Zeta function. Exboço could show me a proof of this where I could find it? ...
3
votes
1answer
172 views

An example of topological space in which each singleton is not in $G_\delta$

Does there exist a compact Hausdorff space such that each singleton is not in $\cal{G_\delta}$? Maybe not difficult example.
7
votes
0answers
586 views

Errata for Dieudonné's Treatise on Analysis volume 2 second edition

I was looking again at the beautiful and quite complete work of Dieudonné, his Treatise of Analysis, to refresh my memory about some aspects of classical analysis. I especially love this Treatise for ...
1
vote
1answer
101 views

Find function if known some limit

I have trouble with the following problem. Let $f=f(p)$, $p>1$, $0<f<1$, and $\lim_{p\to\infty}f(p)=1$. Find $f(p)$, such that $$\lim_{p\to\infty}p\left(1-\sqrt{p ...
3
votes
2answers
112 views

finding maximum of a function on a closed set

I need to find the maximum of the function $\ f(x,y,z) = y $ on the following closed set : $\ y^2+x^2 + z^2 = 3 $ $\ y+ x + z=1$ But I don't have a clue on how to do it ... Trivially ...
0
votes
2answers
76 views

What is $C^{2,1}(\Omega)$ for general $\Omega$?

What is $C^{2,1}(\Omega)$ if $\Omega$ is arbitrary (i.e. neither open nor closed in general)?
9
votes
1answer
190 views

Fourier Series involving the Jacobi Symbol

We know that the Fourier Series $$s(x)=\sum_{k\neq0}\frac{1}{k}\exp\left(2\pi ik x\right)$$ corresponds to the sawtooth function, $s(x)=\left\{x\right\} -\frac{1}{2}$. Suppose that ...
3
votes
1answer
275 views

Norm with special conditions

Let $N$ be a norm on $\mathbb R^2$, such that $N ( \mathbb Z^2) \subset \mathbb N $, where $\mathbb Z^2 =\{ (a,b)\mid a\mbox{ and }b \mbox{ are integers}\}$. Help me to prove that for $u$, $v$ fixed ...
0
votes
3answers
411 views

How can I solve this integral equation using characteristic values and eigenfunctions?

$$ f(x)= \int_0^1 e^{|x-t|} f(t) \, dt+x-1 $$ I can't solve it, because I can't find the boundary conditions?
14
votes
3answers
1k views

$f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$

Trying to solve $f(x)$ is uniformly continuous in the range of $[0, +\infty)$ and $\int_a^\infty f(x)dx $ converges. I need to prove that: $$\lim \limits_{x \to \infty} f(x) = 0$$ Would ...
5
votes
0answers
81 views

surface of a torus by integration [duplicate]

Possible Duplicate: surface area of torus of revolution Let $R>r>0$ fixed. I want to compute the Area of $S=\operatorname{Im} \phi$ given by $$\phi(s,t):= \begin{pmatrix}(R+r\cos s ...
1
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0answers
458 views

How to show that a uniformly continuous function is bounded? [duplicate]

Possible Duplicate: If $f(x)$ is uniformly continuous at $(0,1)$ then is it bounded at $(0,1)$? Uniform continuity and boundedness This was a homework assignment I was asked to do: Let ...
2
votes
1answer
103 views

Integral formula for $f(\sqrt x)$, where $f $ is smooth and even

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be even smooth function. Let $w(x)=f(\sqrt x)$ for $x>0$. On MO http://mathoverflow.net/questions/65264 I found the following integral formula ...