1
vote
0answers
34 views

Rapidly Decreasing Functions

Can someone explain the notion of a rapidly decreasing function? Namely, a function in the Schwartz space: $$\mathscr{S}(\mathbb{R}^n):= \{ f \in C^{\infty} (\mathbb{R}^n) : ||f||_{\alpha, \beta} ...
0
votes
1answer
13 views

What means $df(\tilde x) \in {\mathcal{L}(\mathbb{R}^n)}$

I'm trying to learn math on my own. The bad thing is, I can't google latex letters and they often have multiple meanings. For exmaple ${\mathcal{L}}$ could stand for lagrangian or something else. The ...
1
vote
1answer
28 views

Equivalent Definitions of Negative Order Sobolev Spaces

Ignoring fractional sobolev spaces, if we restrict ourselves to $k>0$ when $k$ is an integer, then the Sobolev space of order $k$, for $W^{k,p}(\mathbb{R})$ is the space of functions $f$ such that ...
2
votes
1answer
22 views

Definition of an integrand

General Question: Say we have integral $$ \int f(z)\ dz $$ Is the integrand in this context (i) $f(z)$ or (ii) $f(z)\ dz$? In any case, is $f(z)\ dz$ a formally defined mathematical object in its ...
6
votes
2answers
210 views

The phrases “has … in ” vs. “contains … of” in Baby Rudin

Consider the following two statements. (Assume $E \subseteq K$.) $E$ has a limit point in $K$. vs. $E$ contains a limit point of $K$. What do they each mean and how are they different?
2
votes
1answer
63 views

The differential is NOT the Jacobi Matrix?

In the book Analysis II by C.T. Michaels the differential is introduced as the Jacobi-Matrix. In class we had the following definition: Definition: Let $U \subset \mathbb{R}^m$ be open, $f: U \to ...
1
vote
1answer
35 views

What can you tell me about integrable functions and riemann integrals?

Define the concepts of integrable function and Riemann integral for functions of two variables (across a rectangle and over an arbitrary area). I know how to define for a rectangle but not an ...
0
votes
1answer
29 views

Analytic, never zero on domain D graph and its local max. min

If $f$ is analytic and never zero on domain D, then $|f(z)|$ has no local minimum or maximum. Now, If I use the fact that $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$), then I can show ...
1
vote
0answers
24 views

Positive definite kernel vs. positive definite function

What is the difference between positive definite kernels and positive definite functions? As I understand it, a positive definite kernel is a positive definite function if it is translation ...
5
votes
6answers
237 views

Why is exponentiation defined as $x^y=e^{\ln(x)\cdot y}$?

There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?
3
votes
2answers
122 views

Radius of Convergence and its application to a Power Series including $x^{2n}$ rather than $x^n$

(Radius of Convergence) Consider the Power Series $f(x)=\sum_{n=0}^{+ \infty}a_n x^n$, the radius of convergence $\rho$ can be found using $$\rho = \displaystyle \lim_{n \to + \infty} \left| ...
0
votes
1answer
169 views

For all but finitely many $n \in \mathbb N$

In my book I have the following theorem: A sequence $\langle a_n \rangle$ converges to a real number $A$ if and only if every neighborhood of $A$ contains $a_n$ for all but finitely many $n \in ...
0
votes
2answers
69 views

Different ways of defining Absolute Value

Calculus I presents this definition of absolute value: $$f(x)=y=|x|=\left\{\begin{array}{}\;\;\;x&\text{ if}\,\,\,x\geq 0\\-x&\text{ if}\,\,\,x<0\end{array}\right.$$ But you can also ...
0
votes
0answers
65 views

Formal and general definition of natural domain (natural set) of a function

Can anyone give me a (as much as possible) formal and general definition of natural domain of a function? Let's say that a function is a triplet $(X,Y,f)$ where $f \subseteq X \times Y$ such that ...
2
votes
0answers
62 views

Clarification of Hölder norm in terms of oscillation

Let $\Omega\subset\mathbb{R}^2$ be an open bounded set, $B(x_0, \rho)=\{x\in\mathbb{R}^2\ |\ |x-x_0|\leq \rho\}$, $\Omega(x_0, \rho)\equiv B(x_0, \rho)\cap \Omega$, $u\in L^{\infty}\big(\Omega(x, ...
3
votes
1answer
93 views

$C^{2, \alpha}$ regularity for elliptic equations with Neumann boundary conditons

Say $\Omega\subseteq \mathbb{R}^n$ is a bounded open set and $0<\alpha<1$. I need some $C^{2, \alpha}(\overline\Omega)$ regularity result for elliptic equations with Neumann boundary conditions ...
3
votes
2answers
611 views

Why do functions with compact support include those that vanish at infinity?

The support of a function is defined in Wikipedia as "the set of points where the function is not zero-valued, or the closure of that set". Functions with compact support in $X$ are defined in ...
2
votes
0answers
69 views

Meaning of quasiperiodicity in classical KAM

I'm learning about the classical KAM theorem, and I can't quite infer precisely what the term "quasi-periodic solution" means in the theorem's statement. I'm reading the following introductory note: ...
1
vote
1answer
87 views

Why this functional isn't differentiable?

I saw in the article Alt, H. M. and Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325, (1981), 105–144. That the functional ...
3
votes
1answer
49 views

What is the meaning of $f(x) \rightarrow a$ as $g(x) \rightarrow b$?

The motivating example was the case: $$f(x, y)\rightarrow0\mathrm{\ \ as\ \ }\sqrt{x^2+y^2}\rightarrow\infty$$ What exactly does this mean? I might define it as: Any sequence $x_n$ with ...
1
vote
2answers
112 views

Definition of $C^k$ boundary

Can someone give me a resonable definition of $C^k$ boundary, e.g., to define and after give a brief explain about the definition. I need this 'cause I'm not understanding what the Evan's book said. ...
1
vote
1answer
51 views

question about Morse theory in Hilbert space

This is sade to be the Morse theory in Hilbert space ,and i want to know the definition (or where i can find it ) of : The qth singular relative homology groupe The qth critical group Please; ...
1
vote
2answers
109 views

Definition of metastability

I was reading Terence Tao's blog post on analogies between soft and hard analysis when I saw that the soft analysis statement "$x_n$ is convergent" corresponds to the hard analysis statement "$x_n$ is ...
3
votes
0answers
50 views

Problem of understanding

Please , can someone help me to understand this part of text please What it means "... for some $k\in \mathbb{N} \cup \lbrace 0 \rbrace$ and uniformly fore a.e $t\in [0,2\pi]$" and please how to ...
1
vote
3answers
141 views

Is a function always a monotonically increasing function

For $\lim_{x \rightarrow a}f(x) = L$, if $f(x)$ is not a constant, is $\delta(\epsilon)$ always a monotonically increasing function? Alternatively, how does the definition of a limit guarantee that ...
2
votes
0answers
227 views

Lemma of Whitehead

this is the lemma of Whitehead And i really don't understand the proof How to see that $k$ is well defined (i.e how to write an element from $X\cup_{\varphi_i}e^{\lambda} , i=0,1$ ) and how to ...
2
votes
1answer
175 views

The definition of $p$ capacity of a set $A\subset\mathbb{R}^n$

I am having a bit of difficult understanding the definition of the $p$-capacity of a set $A\subset\mathbb{R}^n$ and I was wondering if anyone would be able to clarify whether I have the right idea or ...
1
vote
0answers
110 views

How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$

Let $u(x,y)\to l$ as $(x,y)\to z_0=(x_0,y_0).$ How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$ Choose $\epsilon>0.$ Then ...
3
votes
2answers
951 views

Direct sums and direct products

This question has been in my head for a while. And today it appears again when I am reading Arveson's book on $C^*$-algebras. He says Countable direct products of Polish spaces are Polish. ...
1
vote
2answers
78 views

what does it mean to say a space is norm separable?

I came across in my textbook the term: norm separable. I looked in the textbook and online and could not find a definition.
0
votes
0answers
75 views

Definition of matrix derivative

Let $A$ be an $m \times n$ matrix. Let $a_{ij}$ be an element of $A$. What does the notation $\frac{\partial A}{\partial a_{ij}}$ mean?
0
votes
0answers
55 views

Help in Analysis

I'm studying the article "On W1;p Estimates for Elliptic Equations in Divergence Form" of L. A. CAFFARELLI and I. PERAL. There, you can find We use the classical Hardy-Littlewood maximal operator, ...
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?
0
votes
1answer
190 views

Differentiability of a matrix function

To prove the differentiability of a function defined on $M\in M_n$, how should I adapt the definition of differentiability? $h^{-1}[f(x+h)-f(x)-hL(x)]\to 0$  How do I take the reciprocal of a ...
1
vote
2answers
27 views

Definition clarifications on functions of vectors and their derivatives

Definition clarifications would be appreciated: How do I interpret the following ? For $f: R^n\to R^m$, $Df(\vec\xi)(\vec{x})$ in differentiation of a vector function. I know it as a function that ...
0
votes
1answer
97 views

Definition of $\ell_\infty$

What does the $\ell_\infty$ space stand for? Thank you.
1
vote
1answer
150 views

Rephrasing Munkres' Theorem Re: Inverses of Jacobians

Below are theorems from Munkres' "Analysis on Manifolds". The proof of Theorem 7.4 on the right invokes the chain rule, stated on the left. The conditions of Theorem are somewhat strange and appear ...
5
votes
1answer
149 views

What motivates discrepancies between the definitions of “continuous” and “limit”?

I am working from Munkres' Analysis, and I've converted his definitions slightly to make them easier to compare. In the table below, you can fill in the blanks in the top row with words from the ...