For requesting, clarifying, and comparing definitions of mathematical terms.

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0
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6answers
384 views

Orthogonal Projections: Definition + Characterization

Hey there I'm hanging at so many little steps when proving: $P\text{ orthogonal projection}\iff X\text{ orthogonal decomposable}$ My problem is there is simply missing a precise definition of what an ...
0
votes
2answers
55 views

What is the definition of 'line' in $\hat{\mathbb{C}}$?

What is the definition of straight line in $\hat{\mathbb{C}}$? Is it defined as $\{x\in\mathbb{C}: \frac{Re(x-a)}{Re(b)} = \frac{Im(x-a)}{Im(b)}\}\cup \{\infty\}$? ($a,b$ are complex numbers and $b\...
2
votes
2answers
123 views

Odd and even functions.

I have a book which says: If a function $f$ satisfies $f(-x)=f(x)$ for all $x$ in its domain, then $f$ is called an even function. However, if $f(-x)=-f(x)$ for every $x$ in the domain of $f$, ...
1
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3answers
123 views

Orthonormal Bases

I am struggling to get my head around orthonormal bases, this is the defintion in my course notes: If anyone could clarify/explain the concept to me, it would be much appreciated. I am a university ...
1
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2answers
127 views

Rank of a Matrix under certain conditions

I am a little confused about the rank of a matrix. When does the rank of a matrix equals to zero? Is rank of a matrix equal to zero when it is a zero matrix or the matrix has no elements in it? Thank ...
1
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0answers
41 views

Computing the differential (not the Jacobi-Matrix) independent of a basis choice

I am eager to learn more about the differential. I was told that it is a good practice to compute the differential independent of a choice of a basis using the following definition Definition: $f:...
2
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1answer
99 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 ...
0
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1answer
49 views

Derivatives from the left and right

I need to give a definition of the derivative from right and from left. (I know it doesn't make a whole lot of sense, but it is supposed to be similar to the same thing as the limits from the right ...
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2answers
41 views

Homomorphisms, Linear Transformations, and Distributivity

What are the differences between homomorphisms, linear transformations, and distributive operations? To me, they all seem essentially the same, they just are different names for the same phenomenon ...
1
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1answer
32 views

A question about the span of functions.

I am having trouble understanding the span of functions, my problem is: What is $\operatorname{span}\{a\sin(x),b\}$, and what is its dimension? I understand this in terms of vectors, but not in ...
1
vote
1answer
23 views

Can't remember definition of $\lvert G \rvert_{p'}$

For $G$ a finite group, I know that $\lvert G\rvert$ denotes the order of the group. My question is: What is $\lvert G\rvert_{p'}$? Also is this the same as $\lvert G\rvert_p$ (without the prime on ...
0
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1answer
21 views

Writing order of geospacial coordinates

Is there a defined order in which to write geospacial coordinates? When looking at GPX files generated by my GPS, latitude is named first but when looking at a KML file generated by google earth ...
6
votes
2answers
271 views

If a finite poset has greatest and least elements, is it a lattice?

Let $P$ be a finite partially ordered set with elements $0$ and $1$ such that $0 \le x \le 1$ for any $x \in P$. Does it follow that $P$ is a lattice? If not, what is a counterexample? I believe this ...
0
votes
2answers
97 views

What is the definition for holomorphic functions on the Riemann sphere?

I'm trying to study complex analysis in general setting, but i have troubles with defining things in this general setting. I have skimmed definitions in Ahlfors, Conway and wikipedia, but these all ...
4
votes
3answers
549 views

Definition of set.

A set is defined as a collection of distinct objects. Why have we defined a set to contain only distinct objects? Why is a collection of objects which may have identical objects not called a set? ...
3
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1answer
91 views

Equivalent definitions for strictly positive elements

We have two usual definitions for strictly positive elements in C*-algebras: Let $A$ be a C*-algebra Definition (a) [MURPHY, C$^*$-algebras and Operator Theory] An element $a\in A_+$ is said to be ...
0
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4answers
923 views

What is the definition of a positive integer?

I am reading the book "The Number-System of Algebra (2nd edition)". At the starting of page-4 the author writes: A positive integer is a symbol for the number of things in a group of distinct ...
1
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1answer
80 views

What is this metric called?

Ahlfors -complex analysis p.20 Consider a stereographic projection between the 2-sphere and $\overline{\mathbb{C}}$ (i.e. one-point compactification of $\mathbb{C}$) Let $z,w$ be complex numbers. ...
1
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1answer
69 views

A question on Holder spaces

A function $f$ is said to belong to the Holder space if Holder condition is satisfied, i.e. $\exists \beta,L\geq0$ such that $$|f(x)-f(x')|\leq L|x-x'|^\beta$$ for all $x,x'$ in the domain of $f$. I'...
2
votes
1answer
51 views

Why do we define product of morphisms in this way

I'm always forget the definition of product of morphisms in a category, maybe the main reason is because I don't know the motivation beyond the definition: I need help to see this abstract ...
2
votes
3answers
32 views

Can someone explain me how to read members of a set to prove uncountability?

I have trouble in understanding what the elements in the following set are: $$ V = \{f:\mathbb{N}\to \mathbb{N} \mid \text{there is $N_f \in \mathbb{N}$ so that $f(x) \le N_f$ for all $x\in\mathbb{N}$...
1
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1answer
1k views

what is the argument of 0?

When $z\neq 0$, $\arg z$ is defined to be the set $\{\theta \in \mathbb{R} : z=|z|e^{i\theta}\}$. What if $z=0$? Usually does one leave the argument of $0$ undefined? Or is $\arg 0 = \mathbb{R}$?
2
votes
1answer
327 views

Embedding and monomorphism

What is the difference between an embedding and a monomorphism? As far as I can see, most introductory abstract algebra texts treat them as if they are the same, i.e. an injective function from one ...
4
votes
1answer
92 views

How to understand structure groups?

I'm studying fiber bundles and I'm somewhat confused on how Structure Groups appears. The definition of fiber bundle I have is the following: A bundle is a tuple $(E,B,\pi)$ where $E,B$ are ...
0
votes
2answers
114 views

Limits to infinity?

As a part of homework, I was asked What does $\lim_{x\to a} f(x)=\infty$ mean? In an earlier calculus class I was taught that in order for $L=\lim_{x\to a}f(x)$ to exist, we need that $L=\lim_{x\...
13
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5answers
4k views

What precisely is a vacuous truth?

Is there a proper and precise definition that goes something like this? Definition. A statement $S$ is a vacuous truth if ... ...
1
vote
1answer
139 views

$\epsilon$ rules for uniform and pointwise convergence.

Could someone please provide with the $\epsilon$ definitions of uniform and pointwise convergence. I'm trying to really get my head around the differences between them (I do know the differences, but ...
4
votes
3answers
214 views

Can I create my own function like Trigonometric or Exponential

When I want to solve mathematical problems, most of the time I meet the following functions Algebraic like polynomials. Trigonometric like sin(), cos(), tan(), cot(). Logarithmic like log(). ...
2
votes
2answers
170 views

How does the epsilon-delta definition define a limit?

I understand what the epsilon-delta definition is saying in regards to the distance from a point c and the distance from your limit, but I don't understand how this defines a limit. Any help is ...
3
votes
1answer
71 views

Linear function definition

I'm trying to figure out what is linearity and what is a linear function. But the wikipedia page confused me. Firstly it defines as polynomial : $f(x) = ax +c$ Than it defines as linear map: ...
1
vote
4answers
225 views

Vertical bar notation: $\frac{d}{dt}|_{t=0}f(a+tv)=$?

The definition of the directional derivative is $D_vf(a)$=$\left.\dfrac{d}{dt}\right|_{t=0}f(a+tv)=\lim \limits_{t\to0}\dfrac{f(a+tv)-f(a)}{t}$. However, I do not understand how this bar notation is ...
5
votes
4answers
301 views

What is the *exact* definition of a bounded subset in a metric space (in relation with the Heine-Borel Theorem)?

I see quite a lot of different definitions of a bounded space. For instance, from nLab: Let $E$ be a metric space. A subset $B⊆E$ is bounded if there is some real number $r$ such that $d(x,y)<r$...
1
vote
1answer
108 views

A set with a supremum and an infinum inside

What is the name of a set $A$ that has its infimum $i \in A$ and a supremum $s \in A$? Examples: $[0,1]$ has a supremum $1$ and an infimum $0$ which are inside $[0,1]$. $\{0,1,2,3\}$ has a supremum ...
1
vote
4answers
117 views

How is the word “contains” defined in set theory? (In relation with neighborhoods in topology).

From Wiki: Some basic sets of central importance are the empty set (the unique set containing no elements) Thus, this make me think that "contained" is equivalent to the $\in$, as in: if $a$ is ...
2
votes
1answer
583 views

Clear definition of degeneracy of a graph.

There are at least two questions on this topic but the answers are not clear to me and WiKi link didn't make it any clear either. Could someone please clarify is the degeneracy of a graph $G$ ...
1
vote
1answer
266 views

What is the definition of “Winning Strategy” in an Ehrenfeucht-Fraïssé game?

I've read many descriptions and applications of a Winning Strategy, as much as many for a Strategy tout court, but when a formal, algebraic definition is called upon, I've found close to no input. I ...
0
votes
2answers
41 views

Confused about class definition

I find this in my Set theory material: [0] = {x:0==x(mod2)} = {x:2|0-x} , where I'm replacing equivalence sign ("=" with extra horizontal line) with double ...
0
votes
4answers
97 views

Are the axioms of a topological space superfluous?

A topology on a set $X$ is a family $\mathcal{T}$ of subsets of $X$, which are open sets and satisfy: (1) $\emptyset, X \in \mathcal{T}$. (2) Any union of elements of $\mathcal{T}$ belongs to $\...
5
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0answers
93 views

Are the words “function”, “map”, and “mapping” synonymous? [duplicate]

Is it correct to say that "A function or a map or a mapping is a binary relation such that ..."
3
votes
2answers
81 views

Should “together with” be taken as slang for an n-tuple?

When an algebraic structure is defined, it is often defined as a set $S$ "along with"/"together with"/"having" operations $\circ_1, \circ_2, \ldots, \circ_n$, and "denoted" by $(S, \circ_1, \circ_2, \...
1
vote
1answer
60 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
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0answers
41 views

About definition of “extended absolute value” (in $\overline{\Bbb{R}}=\Bbb{R}\cup\{+\infty,-\infty\}$)

Is correct following definition? Def.: Let be $a \in \overline{\Bbb{R}}=\Bbb{R}\cup\{+\infty,-\infty\}$, $$|a|=\begin{cases} |a|& \text{ if } a\in \Bbb{R} \\ +\infty & \text{ if } a \in \{...
0
votes
1answer
48 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
2answers
470 views

Definition of subalgebra

Is it generally implicitly assumed that if $B$ is a subalgebra of a unital Banach algebra $A$ then $1 \in B$? I tried to find a definition of subalgebra but the only definition I found was in Murphy ...
1
vote
1answer
26 views

'Union' of maps

Let $f : A \to Y$, $g : B \to Y$. Suppose that $f(x) = g(x)$ whenever $x \in A \cap B$. Define $$ h : A \cup B \to Y, \\ h(x) = \begin{cases} f(x) & \text{ if $x \in A$} \\ g(x) & \text{ if $...
0
votes
4answers
676 views

Definition of Normalized Number

Which is correct? Are they both correct? Definition 1 A floating point number is called normalized if the leading digit of the fraction is nonzero. for example $(0.10101)_{2}\times 2^{3}$ is ...
0
votes
2answers
61 views

What is the easiest way to prove by induction?

Is there any easy way to do this? I get the basic step.. where you prove it for some number.. but I don't get the induction step. Do you literally take the given equation that you just proved with ...
17
votes
3answers
1k views

What is category theory?

I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is. Is category theory a content in ZFC-set theory? (Just like measure theory, group theory ...
0
votes
1answer
64 views

Limit Point of a Set

Definition. A point $x$ is a limit point of a set A if every $\varepsilon$-neighborhood of $x$ intersects the set A in some point other than $x$. I understand the definition in that $x$ is our limit ...
0
votes
1answer
126 views

Is the value of the sum $1-1+1-1+1-1+\cdots$ does not exist? [duplicate]

Let $a_k:=(-1)^k$ where $k\in\mathbb{N}$. $\mathbb{N}$ is the set of all non-negative integer. And we define the partial sum $S_n:=\sum \limits_{k=0}^{n}a_k$. Notice that the sequence $\{S_k\}$ ...