# Tagged Questions

51 views

### Real Analysis and dynamics

I am looking for a textbook or similar resource that addresses the content in a rigorous graduate course in real analysis(at the level of Rudin/Royden) with the following criteria: No hand waving - ...
83 views

### Any good, undergraduate level introductions to Functional Analysis?

In my lower division math classes, my instructors referenced functional analysis as essentially the extension of linear algebra to infinite dimensional vector spaces along with some real analysis. As ...
10 views

### Interior ball condition in $C^2$ domains

Why a $C^2$ domain satisfies the interior ball condition? I accept a reference too. Thank you.
26 views

### Reference needed for the following sobolev inequalties

I'm reading a paper and the authors applied the following sobolev type estimates $$||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2}$$ for $\alpha>\frac{1}{4}$, where $v$ ...
38 views

### Reference for a Cantor set in the plane formed from series of roots of unity

This is a long shot, but I'm looking for a particular article that I once read, and I'm trying to find it again. It deals with a certain Cantor set in the plane. The set could be written as something ...
24 views

### General solution of the recurrence equation with real shifts

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
123 views

### Smooth spectral decomposition of a matrix

Let $A : x \mapsto A(x)$ be a $C^\infty$ map from the half-plane $\left\{ (x_1,x_2,\cdots,x_n) \in \mathbb{R}^n,\ x_n>0\right\}$ to the space of symmetric matrices with real coefficients. Suppose ...
97 views

### A bounded subset in $\mathbb R^2$ which is “nowhere convex”?

Let $F : \mathbb S^1 \to \mathbb R^2$ represents a simple closed curve $C$ in $\mathbb R^2$. The Jordan curve theorem says that the curves bounds a interior domain $\Omega$ and $\partial \Omega= C$. ...
28 views

389 views

### Proof of Riemann-Lebesgue lemma

I read a book, and this mention to the following lemma of Rieman-Lebesgue type. Lemma. Let $-\infty<a<b<\infty$ and $f(x,y):[a,b]^2\to\mathbb R$ be an integrable and nonnegative function. ...
150 views

### What books on analysis after someone has finished all 3 by Rudin?

What books on analysis would people recommend after someone has finished all three by Rudin (Principles of Mathematical Analysis, Real and Complex Analysis, and Functional Analysis)? I am looking for ...
31 views

### question on isomorphism of abelian von neumann alegbras

I came across the following sentence which I do not know how to prove when reading a paper, "Suppose $(X_0,\mu_0)$ is a non-trivial atomic probability space, then we can identify ...
45 views

### $\sum_{i=1}^{n}\operatorname{Res}(f,z_{i}) + \operatorname{Res}(f,\infty) = 0$

Let $f \in H(\mathbb{C}- \{ z_{1}, \dots, z_{n} \})$. I need a proof of the fact that $$\sum_{i=1}^{n}\operatorname{Res}(f,z_{i}) + \operatorname{Res}(f,\infty) = 0.$$ Where can I find it ?
54 views

### $\prod_{n}f_{n}$ converges uniformly $\Rightarrow$ $\sum_{n}\mathrm{Log}(f_{n})$ converges uniformly

Let $\prod_{n}f_{n}$ be an infinite product of holomorphic functions on a given domain $\Omega$ converging uniformly on compact subsets of $\Omega$ to $f$. Then is it true that ...
52 views

### $2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$

Let $D = B(0,1) \subset \mathbb{C}$ a disc, $f$ holomorphic on $D$. I want to demonstrate that if $$2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$$ then $f$ is linear. I know this is a well-known ...
62 views

### Is Euler's Introductio in analysin infinitorum suitable for studying analysis today?

I've read the following quote on Wanner's Analysis by Its History: ... our students of mathematics would profit much more from a study of Euler's Introductio in analysin infinitorum, rather than ...
47 views

### Relation between the convergence of $\sum a_{n}$ and $\prod (1+a_{n})$ [duplicate]

What is the relation between the convergence of $\sum a_{n}$ and $\prod (1+a_{n})$ where $a_{n} \in \mathbb{C} \ \forall n$ ? Where can I find some references about this topic ?
171 views

### C* Algebra textbook recommendation

I have read the first two chapters from Analysis Now and the chapter on C* algebras (chptr 8?). I'm taking a course on C* algebras in the spring and am currently overwhelmed with the choices. I'd ...
38 views

### Nodes of eigenfunctions and Courant's nodal domain theorem

I am looking for a reference for properties of eigenfunctions of the Laplacian (on the Euclidean plane, and maybe also Laplace-Beltrami on a general manifold): The discreteness of the set of ...
86 views

### Banach Measures: total, finitely-additive, isometry invariant extensions of Lebesgue Measure

I've been reading about paradoxical sets, mainly paradoxical subsets of the plane. As a consequence of this, I've been reading a couple of G.A. Sherman's papers on the subject. In his paper ...
186 views

### How much time is reasonable to complete baby Rudin?

I've been teaching myself math for more than a year. My current aim is towards algebraic topology and differential geometry. Apart from a messy (by which i mean some rigorous and some not) ...
29 views

### Schauder estimate with right hand side in $L^n$.

The classical Schauder estimate says that if $u$ is a solution of $$\Delta u = f$$ where $f \in C^{\alpha}(B_1)$, then $u \in C^{2, \alpha}(B_{1/2})$. Moreover, we have ...
33 views

### How much larger is the $\sigma$-algebra than the algebra in Caratheodory extension?

Given a 'measure' $\lambda$ on an algebra $\mathcal{A}$ of sets, Caratheodory gives a way to extend this $\lambda$ to a $\sigma$-algebra. The idea is we define an outer measure (on all subsets) ...
80 views

### Equality in Minkowski's theorem

I would like to see a proof of when equality holds in Minkowski's inequality. The proof is quite different for when $p=1$ and when $1<p<\infty$. Could someone provide a reference? Thanks!
60 views

### What is the best way to define the diameter of the empty subset of a metric space?

This question is related to Why are metric spaces non-empty? . I think that a metric space should allowed to be empty, and many authorities, including Rudin, agree with me. That way, any subset of a ...
338 views

### Good PDE books for a graduate student?

I am now a graduate student in mathematics, and I really want to learn more about PDE. I would say I have a very solid foundation in soft analysis, including functional analysis and harmonic analysis, ...
1k views

### Functional analysis textbook (or course) with complete solutions to exercises

I am a Ph.D. student in economics and I plan to study functional analysis by myself either this winter or the next summer. I am currently looking for a textbook, and since I am studying it by myself, ...
181 views

### Any suggestions about good Analysis Textbooks that cover the following topics?

I am an undergraduate math major student. I took two courses in Advanced Calculus (Real Analysis): one in Single variable Analysis, and the second in Multivariable Analysis. We basically used Rudin's ...
60 views

### References on the Nash-Moser Implicit Function Theorem

To learn, the Nash-Moser implicit function theorem, I tried with Hamilton (1982) The Inverse Function Theorem of Nash and Moser. But, the article is very encyclopedic. I have a background in ...
96 views

### Suggestions for comprehensive maths book library

I've problem that I'm slowly forgetting the math I've learned in early years at university (right now I'm in final year of Mgr. degree as theoretical physicist). I'd like to assemble a finite but ...
360 views

### An overview of analysis

I'm looking for a book that gives an overview of analysis, a bit like Shafarevich's Basic Notions of Algebra but for analysis. The book I have in mind would give definitions, theorems, examples, and ...
205 views

### From analysis of realvalued functions to analysis of Hilbert/Banach-valued functions

Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ? I'm ...
33 views

57 views

### Measures whose projections are absolutely continuous

Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt rotations) projections of $\mu$ to lines on the plane are absolutely ...
32 views

### existence theorem of eliptic equation

Consider $\Omega \subset R^n$ a bounded and open set with $\partial \Omega$ smooth . Consider the problem: $$- \Delta u + au = f \text{ in } \Omega$$ $$u = 0 \text{ in } \partial \Omega$$ ...
351 views

### Is this reading path recommended?

Since doing math requires learning it first, I 've chosen a series of books to understand some ''Higher math''(which I want to read over a period of several years),and would like to see some ...
68 views

### Reference request: Analysis of real functions on a sphere or torus

I am reading an article in which real functions on the 2-dimensional sphere (in $\mathbb{R}^3$), $f:\mathbb{S}^2 \to \mathbb{R}$, are considered and analyzed. The analysis discusses the spherical ...
332 views

### mathematical analysis by Apostol vs mathematical analysis 1,2 by Zorich [duplicate]

Which of Mathematical analysis by Apostol or Mathematical analysis 1,2 by Zorich is the book to get for self study and why and how is it better?.I mean something readable and exicting and give a bit ...
135 views

### Integrals of matrix functions

I've stumbled across some math I've never really encountered before, and I would love it if someone could provide me with some useful references and texts on it. I'm dealing with integration over the ...
107 views

### Problem of proofs

I've been away from math for a long time ,and while I was trying to relearn it using Courant and Fritz 's booknon calculus,I loved the explanations but I couldn't solve any exercices(they're almost ...
102 views

### Calculus prequisites book

Everytime I try read a calculus textbook I find that my books (serge lang and gelfand's )didn't cover a subject well (like say minimum of a quadratic polynomial) ...I need a recommendation for a ...