1
vote
2answers
81 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 ...
0
votes
0answers
61 views

What does $\dot{\mathbb{R}}$ mean?

I just run into a notation like this: $\dot{\mathbb{R}}$ $-$ any idea what could it mean? I have guesses, but I'd like to be sure. Thanks in advance! [I ran into this notation in the introductory ...
1
vote
1answer
34 views

Doesn't the $p$ depend on $m$

Consider the following text: where $p \in \mathbb Z$. Can you tell me please: doesn't $p$ depend on the $m$? So it is preferable maybe to write $p_m$? Or does $p$ not depend on $m$?
2
votes
1answer
90 views

$C^\omega$ notation for real analytic functions

I've seen the notation $C^\omega$ used for the set of real analytic functions (e.g. on an interval). Where does it come from? What exactly does it mean? What is the reason behind it? Who first used ...
3
votes
1answer
110 views

What is $\mathbb{R}^\mathbb{R}$

I do not know what it is. $\mathbb{R}$ is the set of real numbers. How come $\mathbb{R}\times\mathbb{R}\times \ldots $? Thanks.
2
votes
0answers
29 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
0answers
39 views

Meaning of theta notation in summation.

Just a simple question: What does the big theta notation mean in this equation? $$S(n,p) = \sum_i^ni^p=\Theta(n^{p+1})$$
2
votes
1answer
34 views

$\varepsilon$-neighborhood set notation as $V_{\varepsilon}(a)$ and proof question

Two questions regarding the set $V_{\varepsilon}(a)$. It is defined in the book I am currently using as $$V_{\varepsilon}(a)=\{x\in\mathbb{R}:|x-a|<\varepsilon\}$$ 1). Why the $V$? I suppose ...
0
votes
1answer
56 views

Convention in Riesz representation theorem vs. tempered distribution theory

We are working over the complex field here. Sometimes analysis textbooks say that every continuous linear functional on $L^p$ is integration against some $f \in L^{p'}$ for $p\in (1, \infty)$, rather ...
1
vote
0answers
31 views

Do we need to pay attention to the codomain of a differentiable function?

I came across the following definitions: We call $M\subset \mathbb R^N$ $m$-dimensional $C^k$-submanifold of $\mathbb R^N$ if for all $a\in M$ there is an open neighborhood $U$ of $0$ in $\mathbb ...
0
votes
1answer
42 views

Is there another notation for a characteristic of a domain of a measurable function?

Let $(X,\Sigma,\mu)$ be a measure space and $E\in\Sigma$. In measure theory language, one can prove that $\int_E f d\mu = \int \chi_E f d\mu$. Here, $\chi_E$ is not a function, but it merely means ...
0
votes
1answer
156 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 ...
1
vote
1answer
26 views

Multiplication formula for Lie derivative

Let $U\in\mathbb{R}^n$ be an open set, and let $f_1,f_2\in C^1(U)$. Prove that $$L_v(f_1f_2)=f_1L_vf_2+f_2L_vf_1$$ Suppose $f_1,f_2:U\rightarrow\mathbb{R}$. Let $p\in U$. We have ...
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 ...
0
votes
1answer
87 views

Notation for differentiable

The conditions for the Mean value theorem is that if $f$ is defined on a closed interval $[a,b]$, f is continuous on [a,b] and differentiable on $(a,b).$ Then there exists a $\xi$ in [a,b] such that ...
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 ...
1
vote
2answers
107 views

Meaning of $\{ a,b \}$, and comparison with $(a,b)$

What does $\{a,b\}$ mean in real analysis? I'm also little bit confused about set definition Can you tell me the main difference between $(a,b)$ and $\{a,b\}$? Thank you.
2
votes
0answers
68 views

Is this symbol $\supset\kern-1.7pt\rightarrow$ commonly used in mathematics?

In Multidimensional Real Analysis I by J.J. Duistermaat and J.A.C. Kolk, the symbol $\supset\kern-1.7pt\rightarrow$ is commonly used. For example, $f: A\supset\kern-1.7pt\rightarrow B$ would mean a ...
1
vote
0answers
54 views

Abbreviation for $n$ times differentiable, with $n$th derivative bounded?

Are there convenient abbreviations in use for the following sets? The set of functions which are $n$ times differentiable, with first $n-1$ derivatives continuous (obviously the last part is ...
0
votes
0answers
66 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 ...
3
votes
4answers
100 views

inf of a finite set

I often see the notation $$\inf \{x_1,..,.x_n\}$$ or $$\inf\{s,t\}$$ Is this considered appropriate notation or is it considered better to use $\min$ in these cases?
2
votes
2answers
59 views

Default metrics for $c_0$ and $l^{\infty}$

In my book there is a question like Let $\{a^{(k)}\}$ be a convergent sequence of points in $l^1$. Prove that $\{a^{(k)}\}$ converges in $l^{\infty}$. Now I don't see it mentioned anywhere what ...
1
vote
2answers
93 views

Replacing the value of a function with the value of the limit - is this a standard construction?

Consider a partial function $f : X \rightarrow Y$ where $X$ and $Y$ are topological spaces and $Y$ is Hausdorff. Note that, although the source of $f$ is $X$, the actual domain of $f$ is a (not ...
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: ...
12
votes
7answers
673 views

Does $\,x>0\,$ hint that $\,x\in\mathbb R\,$?

Does $x>0$ suggest that $x\in\mathbb R$? For numbers not in $\,\mathbb R\,$ (e.g. in $\mathbb C\setminus \mathbb R$), their sizes can't be compared. So can I omit "$\,x\in\mathbb R\,$" and ...
5
votes
1answer
89 views

Confusion over calculus notation (differentials/derivatives)

I have read from multiple sources that dy/dx is not to be interpreted as a ratio as the idea of 'dy' and 'dx' themselves will lead to logical difficulties. However, I have seen in many areas (e.g. ...
1
vote
2answers
164 views

What is the meaning of the expression $\liminf f_n$?

I am a little confused as to what $\liminf f_n$ means for a sequence $f_n$ of functions converging to $f$. I can not locate a definition anywhere.
0
votes
0answers
62 views

What is $\check{C}$ ( C with inverted cirumflex)

Context questions: 1 . $\|\cdot\|$ is the norm induced by : $(f,g)=\int f\overline{g}$ (2-norm) $C[a,b]$ is dense in $\check{C}[a,b]$ under the $\|\cdot\|$ norm but not under $\|\cdot\|_\infty$ ...
2
votes
1answer
119 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
44 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 $$||f(x)||_{\infty} = \max_{x\in R^n}|f(x)| $$ ...
0
votes
1answer
99 views

What are the names of $F_{\sigma},G_{\delta}$?

$F_{\sigma}$ denotes a countable union of closed sets and $G_{\delta}$ denotes a countable intersection of open sets. I can see that there is a different use of article for them. For instance, every ...
3
votes
1answer
146 views

What is the difference between these two convergence notations: $f_n \to f$ and $f_n \nearrow f$ or $f_n \searrow f$

I sometimes see this notation for convergence (speaking for functions): $f_n \to f$. And sometimes, I see following: $f_n \nearrow f$ or $f_n \searrow f$. What is the difference between $\to$ and ...
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?
1
vote
1answer
193 views

Name for the real numbers between $0$ and $1$

I see this class of numbers all the time, so I was wondering if there was a special name for it. How to refer to a number $n$ in $\Bbb R$, such that $0<n<1$?
1
vote
1answer
67 views

What is the usual meaning of the notation $ \| f \|_{C^k (A)}$?

If $f \in C^k ( \mathbb R)$ and $A \subset \mathbb R$, what is the usual meaning of the notation below? $$ \| f \|_{C^k (A)} $$
8
votes
5answers
896 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
53 views

Triple sequence

Suppose I consider a triple sequence indexed by $l,m,n$ and I take limits in the order of $l,m,$ and $n$. Then, should I write this sequence as $x_{l,m,n}$ or $x_{n,m,l}$?
2
votes
3answers
556 views

What is the “status” of the rule that multiplication precedes addition.

What is the source, or "status", of the rule that multiplication is performed before addition? Is it a definitive property of $\mathbb R$, a property that can be derived directly from the definition ...
1
vote
4answers
161 views

Meaning of $f:[1,5]\to\mathbb R$

I know $f:[1,5]\to\mathbb R$, means $f$ is a function from $[1,5]$ to $\mathbb R$. I am just abit unclear now on the exact interpretation of "to $\mathbb R$". Is $1\le x\le 5$ the domain? And is ...
2
votes
3answers
343 views

Meaning of $f:[a,b]\to\mathbb{R}$

I was reading this definition and I have always been stuck as to what it means in words. Could someone explain what the function below means in words? $f:[a,b]\rightarrow \mathbb{R}$ is a continuous ...
3
votes
3answers
1k views

Is there a shorthand or symbolic notation for “differentiable” or “continuous”?

In basic calculus an analysis we end up writing the words "continuous" and "differentiable" nearly as often as we use the term "function", yet, while there are plenty of convenient (and even fairly ...
1
vote
1answer
230 views

Simplifying function notation

For example, in the process of proving that $$\left({\frac{f}{g}}\right)'\left({a}\right)= ...
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 ...
2
votes
1answer
154 views

Bound for multi-index sum

I have difficulties in evaluating the multi-index notation in the following context: Let $x \in R^n$ and let $i$ be a multi-index, $i=(i_1, \dots, i_n)$. Now I want to know the bound of the sum ...
0
votes
0answers
199 views

Expression/unknown Notation for Taylor expansion in multivariate case

For a $C^3$-function $f:R^n \to R$, $x \mapsto f(x)$ I have the Taylor series $$ f(x+y)=f(x) + \nabla f (x)^T y + \frac{1}{2} y^T (\nabla^2 f(x)) y + \sum_{|j|=3} \frac{D^j f(z)}{j!} y^{j}$$ with ...
10
votes
2answers
570 views

Formalizing Those Readings of Leibniz Notation that Don't Appeal to Infinitesimals/Differentials

[disclaimer: I've studied a lot of logic but never been good at analysis, so that's the angle I'm coming from below] in my attempt to find a precise version of the 'definitions' usually given when ...
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
163 views

What could the notation $l^\infty(\mathcal{F})$ mean, where $\mathcal{F}$ is a set of measurable functions?

In the book Weak convergence and Empirical Processes, by Aad W. van der Vaart and Jon A. Wellner, on page 81, the notation $l^\infty(\mathcal{F})$ appears, where $\mathcal{F}$ is a set of measurable ...
14
votes
4answers
529 views

Solving (quadratic) equations of iterated functions, such as $f(f(x))=f(x)+x$

In this thread, the question was to find a $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x)) = f(x) + x$$ (which was revealed in the comments to be solved by $f(x) = \varphi x$ where $\varphi$ is ...