Difference between revisions of "Index of notation"

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The unit interval will be assumed when missing
 
The unit interval will be assumed when missing
 
|-
 
|-
| <math>\mathcal{D}_a(\mathbb{R}^n)</math>
+
| <math>D_a(A)</math><br/>Common: <math>D_a(\mathbb{R}^n)</math>
 
|
 
|
 
* Differential Geometry
 
* Differential Geometry
 
* Manifolds
 
* 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}})
+
| Denotes [[Set of all derivations at a point]] - Not to be confused with [[Set of all derivations of a germ]] which is denoted {{M|\mathcal{D}_p(A)}}<br/>
 +
'''Note:''' This is my/Alec's notation for it, as the author<ref>John M Lee - Introduction to smooth manifolds - Second edition</ref> uses {{M|T_p(A)}} - 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]]
 +
|-
 +
| <math>\mathcal{D}_a(A)</math><br/>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 {{M|T_p(A)}}
 
|-
 
|-
 
| <math>\bigudot_i A_i</math>
 
| <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
 
| 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
 
|-
 
|-
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| Same as <math>\mathcal{L}^p</math>
 
| Same as <math>\mathcal{L}^p</math>
 
|-
 
|-
| <math>T_p(\mathbb{R}^n)</math>
+
| <math>T_p(A)</math><br/>Common:<math>T_p(\mathbb{R}^n)</math>
 
|
 
|
 
* Differential Geometry
 
* Differential Geometry
 
* Manifolds
 
* Manifolds
 
| The [[Tangent space|tangent space]] at a point {{M|a}}<br />
 
| 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)}}
+
Sometimes denoted {{M|\mathbb{R}^n_a}} - '''Note:''' sometimes can mean [[Set of all derivations at a point]] which is denoted {{M|D_a(\mathbb{R}^n)}} and not to be confused with <math>\mathcal{D}_a(\mathbb{R}^n)</math> which denotes [[Set of all derivations of germs]]
 
|}
 
|}
  

Revision as of 02:55, 5 April 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.

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
[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 [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

[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]
[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]\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]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 germs

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
  1. John M Lee - Introduction to smooth manifolds - Second edition