Articles of notation

a question on notation for function spaces

If $X$ is some topological space, such as the unit interval $[0,1]$, we can consider the space of all continuous functions from $X$ to $R$. This is a vector subspace of $R^X$ since the sum of any two continuous functions is continuous and scalar multiplication is continuous. Please let me know the notation $R^X$ in […]

The $I$ in the smallest sigma algebra generated by collection of subsets of $\Omega$

In defining a Borel sigma algebra (and if I understand it right) you can depart from the idea that an arbitrary collection of subsets $\mathcal C$ of the sample space $\Omega$, where $\mathcal C$ will end up being intervals of the real line, generates the smallest sigma algebra $\sigma(\mathcal C)$ containing all elements of $\mathcal […]

using the same symbol for dependent variable and function?

Is it wrong to represent a dependent variable and a function using the same symbol? For example, can we write the parametric equations of a curve in xy-plane as $x=x(t)$, $y=y(t)$ where $t$ is the parameter? Also, if someone write the following equation $y=y(t) = t^2$ where $y$ represents the dependent variable and t represents […]

Is there a recommended symbol for “equal by abuse of notation”?

Can anyone suggest a good candidate for a symbol to be used for “equal by abuse of notation“? I can only think of “$\stackrel{\text{def}}{=}$”, but it does not seem to be quite appropriate. For example, in “$m = m\otimes 1$”, are there any suggestions what would fit here better than “$=$”? This is to write […]

What does it mean to “identify” points of a topological space?

I was recently reading about circle rotations (a basic example in dynamical systems) and got confused by some notation. It said consider the unit circle $S^{1} = [0,1]/{\sim}$, where $\sim$ indicates that $0$ and $1$ are identified. What does “identify” mean? and how is the set $[0,1]/{\sim}$ different from the set $[0,1]$? Thanks!

n-tuple Notation

I am reading a paper (to be able to implement the Baum-Welch algorithm in it) and the following notation is defined: $$ [ a_k ]_{k=i}^j ≡ (a_i, a_{i+1}, \ldots , a_j) $$ $$ [a(k)]_{k=i}^j ≡ (a(i), a(i+ 1), \ldots , a( j)) $$ I (think) the first is shorthand of a n-tuple. I guess the […]

What does “IR” mean in linear algebra?

I am new to linear algebra and I was wondering if I could get some help for this question. I understand if it was something like this IR^2 -> IR. I have no idea what IR(=IR^1) means. Could someone please explain? (MathJax version: $\mathbb R^2\to \mathbb R$)

Difference between $R^\infty$ and $R^\omega$

I know $R^\omega$ is the set of functions from $\omega$ to $R$. I would think $R^\infty$ as the limit of $R^n$, but isn’t that $R^\omega$? The seem to be used differently, but I can’t tell exactly how.

Mathematical notation

Is there some generally accepted notation for squarefree, cubefree, etc. numbers? And is there also some notation for squareful, cubeful, etc. numbers?

$j^2 = 1$, but $j \neq \pm 1$; what is $j$?

I was watching a Numberphile video about the favorite number of some mathematicians, and at one point, the creator of MinutePhysics said the following – Similar to the way that $i$ is $\sqrt{-1}$, but what that actually means is that $i^2$ is $-1$, $j^2$ is $+1$, but $j$ is not $1$. Here’s the video with […]