View source for Index of notation

{{Extra Maths}}Ordered symbols are notations which are (likely) to appear as they are given here, for example <math>C([a,b],\mathbb{R})</math> denotes the continuous function on the interval {{M|[a,b]}} that map to {{M|\mathbb{R} }} - this is unlikely to be given any other way because "C" is for continuous. 

==Ordered symbols==
These are ordered by symbols, and then by LaTeX names secondly, for example <math>A</math> comes before <math>\mathbb{A}</math> comes before <math>\mathcal{A}</math>

{| class="wikitable" border="1"
|-
! Expression
! Context
! Details
|-
| <math>\|\cdot\|</math> 
|
* Functional Analysis
* Real Analysis
| Denotes the [[Norm]] of a vector
|-
| <math>\|f\|_{C^k}</math>
|
*Functional Analysis
|This [[Norm]] is defined by <math>\|f\|_{C^k}=\sum^k_{i=0}\sup_{t\in[0,1]}(|f^{(i)}(t)|)</math> - note <math>f^{(i)}</math> is the <math>i^\text{th}</math> derivative.
|-
| <math>\|f\|_{L^p}</math>
|
* Functional Analysis
| <math>\|f\|_{L^p}=\left(\int^1_0|f(t)|^pdt\right)^\frac{1}{p}</math> - it is a [[Norm]] on <math>\mathcal{C}([0,1],\mathbb{R})</math>
|-
| <math>\|f\|_\infty</math>
|
* Functional Analysis
* Real Analysis
| It is a norm on <math>C([a,b],\mathbb{R})</math>, given by <math>\|f\|_\infty=\sup_{x\in[a,b]}(|f(x)|)</math>
|-
| <math>C^\infty</math>
| 
* Differential Geometry
* Manifolds
| That a function has continuous (partial) derivatives of all orders, it is a generalisation of <math>C^k</math> functions
|-
| <math>C^k</math> ''[at {{M|p}}]''
|
* Differential Geometry
* Manifolds
| A function is said to be <math>C^k</math> [at {{M|p}}] if all (partial) derivatives of all orders exist and are continuous [at {{M|p}}]
|-
| <math>C^\infty_p</math>
|
* Differential Geometry
* Manifolds
| <math>C^\infty_p(A)</math> denotes the set of all [[Germ|germs]] of <math>C^\infty</math> functions on {{M|A}} at {{M|p}}<br/>
[[The set of all germs of smooth functions at a point]]
|-
| <math>C^k([a,b],\mathbb{R})</math>
| 
* Functional Analysis
* Real Analysis
| It is the set of all functions <math>:[a,b]\rightarrow\mathbb{R}</math> that are [[Continuous map|continuous]] and have continuous derivatives up to (and including) order <math>k</math><br/>
The unit interval will be assumed when missing
|-
| <math>\mathcal{D}_a(\mathbb{R}^n)</math>
|
* Differential Geometry
* Manifolds
| Denotes [[Set of all derivations at a point]] - sometimes denoted {{M|T_a(\mathbb{R}^n)}} (and such authors will denote the tangent space as {{M|\mathbb{R}^n_a}})
|-
| <math>\bigudot_i A_i</math>
| 
| Makes it explicit that the items in the union (the <math>A_i</math>) are pairwise disjoint, that is for any two their intersection is empty
|-
| <math>\ell^p(\mathbb{F})</math>
|
*Functional Analysis
| The set of all bounded sequences, that is <math>\ell^p(\mathbb{F})=\{(x_1,x_2,...)|x_i\in\mathbb{F},\ \sum^\infty_{i=1}|x_i|^p<\infty\}</math>
|-
| <math>\mathcal{L}^p</math>
|
* Measure Theory
| <math>\mathcal{L}^p(\mu)=\{u:X\rightarrow\mathbb{R}|u\in\mathcal{M},\ \int|u|^pd\mu<\infty\},\ p\in[1,\infty)\subset\mathbb{R}</math><br/>
<math>(X,\mathcal{A},\mu)</math> is a [[Measure space|measure space]]. The class of all [[Measurable function|measurable functions]] for which <math>|f|^p</math> is integrable
|-
| <math>L^p</math>
|
* Measure Theory
| Same as <math>\mathcal{L}^p</math>
|-
| <math>T_p(\mathbb{R}^n)</math>
|
* Differential Geometry
* Manifolds
| The [[Tangent space|tangent space]] at a point {{M|a}}<br />
Sometimes denoted {{M|\mathbb{R}^n_a}} - '''Note:''' sometimes can mean [[Set of all derivations at a point]] which is often denoted {{M|\mathcal{D}_a(\mathbb{R}^n)}}
|}

==Unordered symbols==

{| class="wikitable" border="1"
|-
! Expression
! Context
! Details
|-
| <math>\mathcal{A}/\mathcal{B}</math>-measurable
|
* Measure Theory
| There exists a [[Measurable map]] between the [[Sigma-algebra|{{sigma|algebras}}]]
|-
| {{M|a\cdot b}}
| 
* Anything with vectors
| [[Vector dot product]]
|}




[[Category:Subjects]]