0
votes
0answers
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 - ...
3
votes
2answers
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 ...
0
votes
1answer
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.
1
vote
0answers
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 $v$ ...
5
votes
1answer
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 ...
0
votes
0answers
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 ...
3
votes
2answers
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 ...
5
votes
1answer
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 $\partial \Omega= C$. ...
0
votes
0answers
28 views

Generalised Holder inequality

I am looking for a theorem of the form If $p\le q$ and $f\colon X\times Y\to \mathbb{\mathbb{R}_+}$ satisfies (assumptions), then $$(\int_X dx(\int_Y dy (f(x,y))^p)^{\frac qp})^{\frac 1q}\le ...
0
votes
1answer
22 views

A characterization of differentiability of a convex function

Let $\phi : \mathbb R^n \to \mathbb R$ be a convex function. For all point $x\in \mathbb R^n$, define the subdifferential as $$\partial \phi(x) = \{ y\in \mathbb R^n | \ \phi(z) \geq \phi(x) + ...
2
votes
4answers
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. ...
1
vote
5answers
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 ...
1
vote
1answer
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 ...
2
votes
1answer
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 ?
1
vote
0answers
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 ...
1
vote
1answer
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 ...
3
votes
0answers
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 ...
3
votes
2answers
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 ?
2
votes
3answers
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 ...
1
vote
2answers
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 ...
0
votes
1answer
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 ...
7
votes
0answers
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) ...
3
votes
0answers
29 views

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

The classical Schauder estimate says that if $u$ is a solution of \begin{equation} \Delta u = f \end{equation} where $f \in C^{\alpha}(B_1)$, then $u \in C^{2, \alpha}(B_{1/2})$. Moreover, we have ...
4
votes
0answers
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) ...
0
votes
1answer
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!
1
vote
1answer
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 ...
6
votes
6answers
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, ...
7
votes
3answers
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, ...
2
votes
3answers
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 ...
6
votes
0answers
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 ...
1
vote
1answer
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 ...
10
votes
6answers
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 ...
1
vote
3answers
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 ...
0
votes
1answer
33 views

Existence of a geodesic in a complete separable metric space

If I have $X$ a complete separable metric space, $x, y \in X$ arbitrary points, how can I define a constant speed geodesic, i.e. a continuous map $g : [0,1] \rightarrow X$ such that $$ d(g(t), g(s)) = ...
1
vote
1answer
136 views

Comprehensive references on partial differential equations

How do the three volumes by Taylor's "Partial differential equations" compare with the two volumes with the same title by Friedrich Sauvigny's as a reference for study? What are the good and bad ...
0
votes
0answers
25 views

Reference for Expansions of elliptic integrals

You can typically write Elliptic integrals in terms of a series expansion. These for example occur when you want to write the period for small oscillations of the (nonlinear) pendulum. I am looking ...
1
vote
1answer
56 views

Where to find a proof of Silverman-Toeplitz?

I am referring to the theorem which gives a necessary and sufficient condition on a infinite matrix that maps convergent sequences to sequences converging to the same limits. Wiki gives a link to ...
1
vote
3answers
97 views

Complex book suggestions

I take complex analysis course. And my instructor use -Bak and Newman's complex analysis book, springer. This book explains too fast and superficially. Please give me book suggestions which are the ...
0
votes
0answers
45 views

Exercise references

I could recommend any good text analysis, or perhaps a list of exercises with good problems (for show) on dips, submersiones and implicit functions. I appreciate any references.
2
votes
0answers
25 views

Weyl asymptotic law

In Panoramic view in Riemannian geometry of Berger, I met the following formula $$\sum e^{-\lambda_i t} \sim \frac{\vert \Omega\vert }{2\pi t} -\frac{\vert \partial \Omega\vert}{\sqrt{2\pi t}} + ...
2
votes
0answers
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 ...
3
votes
0answers
32 views

existence theorem of eliptic equation

Consider $\Omega \subset R^n$ a bounded and open set with $\partial \Omega$ smooth . Consider the problem: $$ - \Delta u + au = f \text{ in } \Omega $$ $$ u = 0 \text{ in } \partial \Omega $$ ...
3
votes
1answer
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 ...
0
votes
2answers
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 ...
0
votes
0answers
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 ...
6
votes
1answer
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 ...
2
votes
1answer
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 ...
1
vote
1answer
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 ...
6
votes
1answer
254 views

Self Study of Fractals

I am looking for a book to self-study fractals with a certain criteria. I have checked out Getting Aquainted with Fractals. Note that Getting Aquainted with Fractals does not include ...
2
votes
1answer
377 views

Is there an English translation of Jordan's “Cours D'analyse”

I am trying to find an English translation of Camille Jordan's work "Cours D'analyse". Only the French edition is on Amazon, so since this is a somewhat specialized topic, I thought perhaps someone in ...