Q: Notation for Differentiable Functions

  • Thread starter Thread starter rudders93
  • Start date Start date
  • Tags Tags
    Notation Space
rudders93
Messages
46
Reaction score
0
Hi,

I was wondering, how can i express the following in notation (function notation i think it is? The one where we {(x,y) \in R3 : x + y = 0}

Q: Set of real polynomials of any degree

Q: Set of all differentiable functions (which I guess just means continuous functions, but nevertheless not sure how to express this properly?)

Thanks!
 
Physics news on Phys.org
Hi rudders93,
I hope the following is what you are looking for (and free from errors)

The space of polynomials in a formal variable x over the field F is denoted
F[x] = { a0 + a1x +...+ anxn | aiF, n < ∞ }
See http://en.wikipedia.org/wiki/Examples_of_vector_spaces#Polynomial_vector_spaces" on Wikipedia for more info.

Continuous does not mean the same as differentiable.
The set of real functions for which the nth derivative exists over the entire domain Ω⊆R is denoted
Cn(Ω) = { f:Ω→R | f'∈Cn-1(Ω) }
This is a recursive definition terminating with continuous functions
C0(Ω) = { f:Ω→R | limx→cf(x)=f(c) ∀ c∈Ω }
For more info see Wikipedia:
http://en.wikipedia.org/wiki/Smooth_function#Differentiability_classes"
http://en.wikipedia.org/wiki/Continuous_function"
 
Last edited by a moderator:
Thanks!
 
##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
Back
Top