Difference between revisions of "Index of notation"

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Due to the frequency of some things (like for example ''norms'') they have been moved to their own index.  
 
Due to the frequency of some things (like for example ''norms'') they have been moved to their own index.  
 
{| class="wikitable" border="1"
 
{| class="wikitable" border="1"
 +
! colspan="4" | Symbols
 
|-
 
|-
 
! Index
 
! Index
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| [[Absolute value]]
 
| [[Absolute value]]
 
| Not to be confused with <math>\Vert\cdot\Vert</math>-like expressions, see above of [[Index of norms and absolute values#Norms|this index]]
 
| Not to be confused with <math>\Vert\cdot\Vert</math>-like expressions, see above of [[Index of norms and absolute values#Norms|this index]]
 +
|-
 +
! colspan="4" | Alphabetical
 +
|-
 +
! Index
 +
! Expressions
 +
! Name
 +
! Notes
 
|}
 
|}
 +
 +
==Index==
 +
Index example: <code>R_bb</code> means this is indexed under R, then _, then "bb" (lowercase indicates this is special, in this case it is blackboard and indicates <math>\mathbb{R}</math>), <code>R_bb_N</code> is the index for <math>\mathbb{R}^n</math>
 +
{| class="wikitable" border="1"
 +
|-
 +
! Expression
 +
! Index
 +
! Context
 +
! Details
 +
|-
 +
| {{M|\mathbb{R} }}
 +
| R_bb
 +
|
 +
* Everywhere
 +
| Denotes the set of [[Real numbers]]
 +
|-
 +
| {{M|\mathbb{S}^n}}
 +
| S_bb_N
 +
|
 +
* Everywhere
 +
| <math>\mathbb{S}^n\subset\mathbb{R}^{n+1}</math> and is the [[Sphere|{{n|sphere}}]], examples:<br/>
 +
{{M|\mathbb{S}^1}} is a circle, {{M|\mathbb{S}^2}} is a sphere, {{M|\mathbb{S}^0}} is simply two points.
 +
|}
 +
==Old stuff==
  
 
==Markings==
 
==Markings==
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! Details
 
! Details
 
! Mark
 
! Mark
−
|-
 
−
| <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>
 
| <math>C^\infty</math>
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* Measure Theory
 
* Measure Theory
 
| Same as <math>\mathcal{L}^p</math>
 
| Same as <math>\mathcal{L}^p</math>
−
|
 
−
|-
 
−
| <math>\mathbb{S}^n</math>
 
−
|
 
−
*Real Analysis
 
−
*Differential Geometry
 
−
*Manifolds
 
−
*Topology
 
−
| <math>\mathbb{S}^n\subset\mathbb{R}^{n+1}</math> and is the [[Sphere|{{n|sphere}}]], examples:<br/>
 
−
{{M|\mathbb{S}^1}} is a circle, {{M|\mathbb{S}^2}} is a sphere, {{M|\mathbb{S}^0}} is simply two points.
 
 
|
 
|
 
|-
 
|-

Revision as of 09:28, 9 June 2015

[math]\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}[/math][math]\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }[/math]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 [ilmath][a,b][/ilmath] that map to [ilmath]\mathbb{R} [/ilmath] - this is unlikely to be given any other way because "C" is for continuous.

Sub-indices

Due to the frequency of some things (like for example norms) they have been moved to their own index.

Symbols
Index Expressions Name Notes
[ilmath]\Vert\cdot\Vert[/ilmath] index Something like [math]\Vert\cdot\Vert[/math] Norm Not to be confused with [math]\vert\cdot\vert[/math]-like expressions, see below or this index
[ilmath]\vert\cdot\vert[/ilmath] index Something like [math]\vert\cdot\vert[/math] Absolute value Not to be confused with [math]\Vert\cdot\Vert[/math]-like expressions, see above of this index
Alphabetical
Index Expressions Name Notes

Index

Index example: R_bb means this is indexed under R, then _, then "bb" (lowercase indicates this is special, in this case it is blackboard and indicates [math]\mathbb{R}[/math]), R_bb_N is the index for [math]\mathbb{R}^n[/math]

Expression Index Context Details
[ilmath]\mathbb{R} [/ilmath] R_bb
  • Everywhere
 Denotes the set of Real numbers
[ilmath]\mathbb{S}^n[/ilmath] S_bb_N
  • Everywhere
 [math]\mathbb{S}^n\subset\mathbb{R}^{n+1}[/math] and is the [ilmath]n[/ilmath]-sphere, examples:

[ilmath]\mathbb{S}^1[/ilmath] is a circle, [ilmath]\mathbb{S}^2[/ilmath] is a sphere, [ilmath]\mathbb{S}^0[/ilmath] is simply two points.

Old stuff

Markings

To make editing easier (and allow it to be done in stages) a mark column has been added

Marking Meaning
TANGENT Tangent space overhall is being done, it marks the "legacy" things that need to be removed - but only after what they link to has been updated and whatnot
TANGENT_NEW New tangent space markings that are consistent with the updates

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]

Expression Context Details Mark
[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

See also Smooth function and the symbols [ilmath]C^\infty(\mathbb{R}^n)[/ilmath] and [ilmath]C^\infty(M)[/ilmath] where [ilmath]M[/ilmath] is a Smooth manifold

[math]C^\infty(\mathbb{R}^n)[/math]
  • Differential Geometry
  • Manifolds
 The set of all Smooth functions on [ilmath]\mathbb{R}^n[/ilmath] - see Smooth function, it means [ilmath]f:\mathbb{R}^n\rightarrow\mathbb{R} [/ilmath] is Smooth in the usual sense - all partial derivatives of all orders are continuous. TANGENT_NEW
[math]C^\infty(M)[/math]
  • Differential Geometry
  • Manifolds
 The set of all Smooth functions on the Smooth manifold [ilmath]M[/ilmath] - see Smooth function, it means [ilmath]f:M\rightarrow\mathbb{R} [/ilmath] is smooth in the sense defined on Smooth function TANGENT_NEW
[math]C^k[/math] [at [ilmath]p[/ilmath]]
  • Differential Geometry
  • Manifolds
 A function is said to be [math]C^k[/math] [at [ilmath]p[/ilmath]] if all (partial) derivatives of all orders exist and are continuous [at [ilmath]p[/ilmath]]
[math]C^\infty_p[/math]
  • Differential Geometry
  • Manifolds
 [math]C^\infty_p(A)[/math] denotes the set of all germs of [math]C^\infty[/math] functions on [ilmath]A[/ilmath] at [ilmath]p[/ilmath]

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 and have continuous derivatives up to (and including) order [math]k[/math]

The unit interval will be assumed when missing

[math]D_a(A)[/math]
Common: [math]D_a(\mathbb{R}^n)[/math]
  • Differential Geometry
  • Manifolds
 Denotes Set of all derivations at a point - Not to be confused with Set of all derivations of a germ which is denoted [ilmath]\mathcal{D}_p(A)[/ilmath]

Note: This is my/Alec's notation for it, as the author[1] uses [ilmath]T_p(A)[/ilmath] - which looks like Tangent space - the letter T is too misleading to allow this, and a lot of other books use T for Tangent space

TANGENT
[math]\mathcal{D}_a(A)[/math]
Common: [math]\mathcal{D}_a(\mathbb{R}^n)[/math]
  • Differential Geometry
  • Manifolds
 Denotes Set of all derivations of a germ - Not to be confused with Set of all derivations at a point which is sometimes denoted [ilmath]T_p(A)[/ilmath] TANGENT
[math]\bigudot_i A_i[/math]
  • Measure Theory
 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]G_p(\mathbb{R}^n)[/math]
  • Differential Geometry
  • Manifolds
 The geometric tangent space - see Geometric Tangent Space TANGENT_NEW
[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]

[math](X,\mathcal{A},\mu)[/math] is a measure space. The class of all measurable functions for which [math]|f|^p[/math] is integrable

[math]\mathcal{L}(V,W)[/math]
  • Linear Algebra
 The set of all linear maps from a vector space [ilmath]V[/ilmath] (over a field [ilmath]F[/ilmath]) and another vector space [ilmath]W[/ilmath] also over [ilmath]F[/ilmath]. It is a vector space itself.

See The vector space of all maps between vector spaces

[math]\mathcal{L}(V)[/math]
  • Linear algebra
 Short hand for [math]\mathcal{L}(V,V)[/math] (see above).

In addition to being a vector space it is also an Algebra

[math]L^p[/math]
  • Measure Theory
 Same as [math]\mathcal{L}^p[/math]
[math]T_p(A)[/math]
Common:[math]T_p(\mathbb{R}^n)[/math]
  • Differential Geometry
  • Manifolds
 The tangent space at a point [ilmath]a[/ilmath]

Sometimes denoted [ilmath]\mathbb{R}^n_a[/ilmath] - Note: sometimes can mean Set of all derivations at a point which is denoted [ilmath]D_a(\mathbb{R}^n)[/ilmath] and not to be confused with [math]\mathcal{D}_a(\mathbb{R}^n)[/math] which denotes Set of all derivations of a germ

TANGENT

Unordered symbols

Expression Context Details
[math]\mathcal{A}/\mathcal{B}[/math]-measurable
  • Measure Theory
 There exists a Measurable map between the [ilmath]\sigma[/ilmath]-algebras
[ilmath]a\cdot b[/ilmath]
  • Anything with vectors
 Vector dot product
[math]p_0\simeq p_1\text{ rel}\{0,1\}[/math]
  • Topology
 See Homotopic paths
  1. ↑ John M Lee - Introduction to smooth manifolds - Second edition
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