# Tagged Questions

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

129 views

### Why so many 'multi-part' definitions, as opposed to 'unified' ones?

Many definitions consist of multiple parts: an equivalence relation is symmetric AND reflexive AND transitive; a topology is closed over finite intersections AND over arbitrary unions; etc. However, ...
27 views

### Understanding the definition of the direct sum of subspaces of a vector space

I have a question regarding the definition of direct sum of a vector space in relation to subspaces. Definition: A vector space $V$ is called the direct sum of $W_1$ and $W_2$ if $W_1$ and $W_2$ ...
41 views

### What is the name of the sub-matrix?

Given a matrix $\mathbf{A}$ of size $n\times n$. Let $I=\{i_1,\ldots,i_k\}\subseteq\{1,\ldots,n\}$ for some $k\leqslant n$. How to call the sub-matrix of $\mathbf{A}$ that has its indices in $I$? (I ...
37 views

20 views

### Given a basis $U$, what conditions are needed for an orthogonal basis for it?

Given a basis $U$, what conditions are needed for an orthogonal basis for it? For example, in the following vector space $U$, if $U =sp\{(1,1,1),(1,3,7)\}$ then what conditions are needed for an ...
99 views

### Codomains and the definition of a function

A function $f$ is defined as a set of ordered pairs $(x, y)$ such that $(x,b), (x,c) \in f \Longrightarrow b=c$. Since $y$ is determined uniquely by $x$, it is customarily denoted $f(x)$. One ...
26 views

### Alternative version of definition of convergence

Knowing the definition of convergence of a sequence in $\Bbb R$: ($x_n$) converges to x iff $\forall \epsilon \gt 0$ $\exists N$ such that for every $n\gt N$, $d(x_n,x)\lt \epsilon$. Consider a new ...
32 views

### Heine definition of limit of a function at infinity using sequences

I couldn't find the answer neither on Google, nor this website, so decided to ask. The Heine definition of limit: from Wikipedia $\lim_{x\to a}f(x)=L$ if and only if for all sequences $x_n$ (with ...
55 views

35 views

183 views

### Why *all* $\epsilon > 0$, in the $\varepsilon-\delta$ limit definition?

Definition of $\lim_{x \to a} f(x) = L$: $\forall \epsilon > 0, \exists \delta > 0 s.t. |f(x) - L| < \epsilon$ $if \ 0 < |x-a| < \delta$ Question: Why can't we weaken the ...
137 views

### Book on foundational reasoning of standard arithmetics “curriculum”

I am interested in a book that is about arithmetics but the presentation is not just the known to all formulas but the foundational logic behind it. The closest example I can think about is the way ...
86 views

49 views

55 views

### Mathematical usage of “$\dots$” during enumeration, is it ok to be imprecise?

I am guilty of writing things like the following in proofs: so by lemma 1.2 we have that for $k<n$ all of the integers $k,k+1,k+2,k+3,\dots ,n$ are pompous. I really like how this looks, the ...
25 views

### Local maxima and minima of functions on a discrete domain

Could we say every point of functions on a discrete domain is local maximum and local minimum? (as long as we are not concerned with strict maximum and minimum) For example, if we we define a ...
38 views

### Finding the probability space of the given experiment.

Specify the probability space completely for the following experiment: tossing a fair coin till we see the first heads. Here is what I have done so far: The sample space is simply $T^n H$ where $n$ ...
26 views

### What does “modulo equivalence relationship” mean?

I am reading something on completion of metric spaces and it says: Let $\hat S$ be $\mathcal{C}$ modulo equivalence relationship of co-Cauchy sequences. Where $\mathcal{C}$ is the set is all ...
37 views

### Definition of mathematical expression

According to wikipedia: "In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical ...
26 views

### Understanding the definition of an Interface

Im starting to learn about modeling of moving interfaces and am feeling daft about the basic definition itself: Given an $n$ dimensional space $\Omega$ an interface $\Gamma$ is a co-dimension 1 ...
25 views

67 views

### Defining vertical tangent lines

In looking at the definition of vertical tangent lines in some popular calculus texts, I noticed that there are a few different definitions for this term, including the following: A function $f$ ...
100 views

### Why the terminology “global fields” and “local fields”

Let $p$ be a prime number. A global field is defined as a finite extension of $\mathbb Q$ or $\mathbb F_p(t)$. On the other hand one can show that a local field (which is by definition a complete ...
41 views

### What is the precise relationship between vector space and vecctor field?

I have looked up precise definition of a vector field and a vector space but I could not understand the relationship between them. On wikipedia A vector field is: Given a subset $S$ in $R^n$, a ...