1
vote
1answer
84 views

What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$?

What is the difference between a sequence of functions $(f_n)$ and a sequence of functions $f_n(x)$? I am reading my textbook on analysis, and it seems to use 'sequence of functions' to describe both ...
4
votes
2answers
101 views

What does this $\asymp$ symbol mean? (subject: analytic number theory)

I'm reading a survey article by Andrew Granville on analytic number theory. On page 22 of the paper, there appears a strange looking symbol, undefined. I've circled it in red in the screenshot ...
2
votes
0answers
31 views

Notation for near optimal solution

Usually, $x^*$ is used to denote the optimal solution to a maximization problem. I need a notation to describe a solution that is not optimal but "good enough." In my case this solution is the first ...
0
votes
1answer
81 views

Notation Question (Meaning of double inclusion symbols)

What does the notation $\subset \subset$ mean? In my class notes, our prof writes $\Omega \subset \subset \mathbb{R}^{n}$ to mean that "$\Omega$ is a convex subset of $\mathbb{R}^{n}$". Is that all ...
0
votes
2answers
52 views

Sequence Notation in Analysis

If real sequence is a function from the set $\mathbb N$ to the set $\mathbb R$ and function is represented by $(a,b)$, where $a$ is domain and $b$ is range, then why do we represent sequence only by ...
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 ...
2
votes
1answer
53 views

Defintion of $\ell^\infty$

I have come across the space of bounded sequences denoted as $\ell^\infty$ in my course, but not a clear, concise definition. I have seen sometimes when these includes sequences in $\mathbb{R}$ that ...
1
vote
1answer
104 views

Problem with notation: Laplacian on a manifold

In the Aubin's book "Nonlinear analysis on manifolds" the Laplacian operator on functions on some smooth manifold is defined by the formula $$ \Delta = -\nabla^\gamma\nabla_\gamma, $$ where ...
0
votes
0answers
51 views

Correct notation in this case

I want to define $y_n:=(1,\frac{1}{2},...,\frac{1}{n},0,..)$(hence, a sequence) for all $n \in \mathbb{N}$. And I was wondering whether $y_n:= \sum_{i=1}^{n} \frac{1}{i} (\delta_{ip})_{p \in ...
0
votes
1answer
50 views

Question about notation in differential equations.

In general, an ordinary differential equation is in the form $$ \begin{cases} x'(t) = f(t, x(t)) \\ x(t_0) = x_0 \end{cases}. $$ When proving the existence and uniqueness theorems, an operator $T$ was ...
0
votes
0answers
68 views

Small symbols behind parantheses

I am currently reading the following paper where the author uses constantly small symbols after parantheses, but I do not know what this means. I am particularly interested in equation (23), so you ...
0
votes
1answer
142 views

Asymptotic notation meaning in transitive relation

I'm attempting to prove the transitive relation on $\theta$ and I'm having trouble understanding the meaning of one of the symbols used. Here is the transitive relation: $f(n) = \theta(g(n)) ...
0
votes
1answer
149 views

How to write the expression with multi-index notation?

I need some help for writing the following expression with multi-index notation, $$\sum_{i_1, \ldots, i_p=1}^n \frac{\partial^{2p}}{\partial x_{i_1}^2\ldots \partial x_{i_p}^2}f(x, \xi),$$ where ...
2
votes
0answers
92 views

Symbol for functions that vanish on boundary?

If I have a domain $ M \subset \mathbb{R}^n $, is there a standard symbol for the set of functions $ f \in C^\infty(M) $ that vanish on $ \partial M$ ? I feel like I have seen this before, but I'm ...
1
vote
0answers
34 views

What's the need of $^{S}_{T}$ in $f^{S}_{T}:S\rightarrow T$?

I'm reading Lang's Undergraduate Analysis: In the chapter about mappings, he says that we should denote the set of arrival and the set of departure with the following notation: ...
1
vote
1answer
25 views

Asymptotic Approximation and Sign Convention

When I write the asymptotic approximation of a function, does the sign convention matter? i.e. suppose I have (though the formula might not make sense) $$f_n(x)=x^2+\dots-O(n),$$ If my function is ...
0
votes
2answers
45 views

Inner product convention for $\ell^p$?

So I'm reading through some analysis problems and one is discussing $\ell^p$ (the space of $p$-summable sequences $x: \mathbb Z^+ \to \mathbb C$ such that $\sum_{n \in \mathbb Z^+}|x_n|^p < ...
2
votes
1answer
85 views

Notation Clarification of Koch Curve

I am having trouble making sense of the notation used to describe the Koch Curve in the book Getting Aquanted with Fractals. The link will take you to a preview of the book which describes the ...
1
vote
1answer
48 views

What does $C^{\text{1,stw}}(0,b)$ mean

I know that this C with the one means continoulsy differentiable functions but what does this stw stand for? Does anyone know this?
2
votes
1answer
123 views

Product rule smart notation

Imagine we have a product of functions $f_1\cdots f_m$. We know a rule to compute the derivative. On the other hand, we also have a rule or formula to compute the $n$-th derivative of $fg$ but my ...
1
vote
2answers
1k views

What does $C[0,1]$ mean?

In the context of real analysis, I have found this question: For each $$f \in C[0,1] $$ there is a series of even polynomials , which converge uniformly on $[0,1]$ to f. What is $C[0,1]$ ? Is it ...
0
votes
1answer
45 views

Given a function space with a norm , what is the meaning of writing $||.||$ when the used norm is $||.||_\infty$

Example 1 Given $$C_{0}(\mathbb{R}^{n})=\{f\in C(\mathbb{R^n} \ | \ \ \exists R \ge 0 \ \text{such that } f(x)=0 \ \text{for} \ ||x||\ge R \}$$ and $$||f(x)||_{\infty} = \max_{x\in R^n}|f(x)| $$ ...
2
votes
4answers
132 views

Understanding Set Notation

I'm having some trouble understanding a definition and explanation in my textbook Introduction to Analysis by Edward Gaughan 5th edition. The book begins with some preliminary information about sets ...
1
vote
1answer
66 views

Nowhere dense notation confusion

The text Foundations of Mathematical Analysis by Johnsonbaugh and Pfaffenberger defines nowhere dense as $X$ is nowhere dense in $M$ if $X^{-,-} = M$. What does this mean?
0
votes
2answers
105 views

How do I read this?

I received the following equation. It is supposed to contain clues to something I have to solve. I am not familiar with the math symbols used here. How do I read the following: ...
1
vote
1answer
1k views

Weighted average vs. weighted mean

Is there a formal difference between weighted average and weighted mean? I get corrected to the latter if I type in the former in wikipedia, and then there is a lot of stuff about the name "average" ...
1
vote
1answer
99 views

Notation question (shorthand for dot products?)

I am reading a paper and trying to understand a calculation and all of a sudden I bump into the following term: $D^2_y p(t,y(t,x))(\partial_t y(t,x),y^\epsilon(t,x))$ where $p$ is a scalar field ...
0
votes
1answer
149 views

indicator function?

Does anyone know what $ 1_\omega v $ means where $v \in L^2((0,T) \times \Omega )$ and $\omega \subset \subset \Omega$? It should be an indicator function of $(t,x)$, but not sure how to interpret ...
8
votes
5answers
918 views

use of $\sum $ for uncountable indexing set

I was wondering whether it makes sense to use the $\sum $ notation for uncountable indexing sets. For example it seems to me it would not make sense to say $$ \sum_{a \in A} a \quad \text{where A is ...
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)?
1
vote
0answers
291 views

Notation for limit points of a minimizing sequence: $\arg \inf$

Could you tell me what is the accepted notation for the set of limit points of a minimizing sequence. For example, if I have a function $f(x)$ and a sequence $x_t$ such that $\lim f(x_t) = \inf ...
2
votes
4answers
266 views

Union of Uncountably Infinite Sets

How does one notationally describe the set which is the union of uncountably many other sets. For instance, for each x such that a < x < b, where a and b are real numbers, if there is assigned ...
1
vote
3answers
284 views

What is the “limit point of a function?”

I am asked to prove that Show that a sequence $x: \mathbb{N} \to \mathbb{R}$ has a limit point iff there $\lim_{n\to\infty} x(n)$ exists as a limit point of a function from a subset of metric ...
1
vote
1answer
161 views

Why is it wrong to express $\mathop{\lim}\limits_{x \to \infty}x\sin x$ as $k\mathop{\lim}\limits_{x \to \infty}x$; $\lvert k \rvert \le 1$?

Why is it wrong to write $$\mathop{\lim}\limits_{x \to \infty}x\left(\frac{1}{x}\sin x-1+\frac{1}{x}\right)=(0k-1+0)\cdot\mathop{\lim}\limits_{x \to \infty}x,$$ where $\lvert k \rvert \le 1$? And, as ...
0
votes
1answer
71 views

Confusion on a differential notation

This is a notation I see in page 8 of Guy Barles and Espen R. Jakobsen, namely $$ \partial_t^{\beta_0}D^{\beta'}\phi(x,t) $$ where $\phi: \mathbb{R}^n\times[0,T]\longrightarrow \mathbb{R}$ is ...
12
votes
3answers
406 views

Uses for esoteric integral symbols

A while ago, I was searching for a TeX package which would provide a double integral symbol with a circle which I could use to typeset some lecture notes on surface integrals. I happened upon the ...
2
votes
1answer
279 views

A question regarding the meaning of “lim”

I'm having an argument about what the notation of $\lim$ means. Assume you have $f_n: X \rightarrow \mathbb{R}$. Are the following two sets equal: $$\{ x \ |\  (f_n(x)) \ \text{converges} \} = ...
2
votes
1answer
108 views

Meaning of $C(I,\mathbb{R})$ and $C^{\infty}(I, \mathbb{R})$ related to continuous functions

What does this mean: ($f$ a function, $I$ an interval and $R$ the real numbers) $f \in C(I,R)$ Does it mean $f$ is an element of the collection of continuous functions with domain $I$ and range $R$ ...
2
votes
2answers
82 views

Newbie question about the re-application of a function on its result

This is a newbie question, but I would be grateful for any reference that you could give. let $f(x) \in \mathbb{A}$, where $x \in \mathbb{A}$ Is there a symbol to indicate the repeated application ...