Difference between revisions of "Index of notation"

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{{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.  
 
{{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.  
 +
 +
==Markings==
 +
To make editing easier (and allow it to be done in stages) a mark column has been added
 +
{| class="wikitable" border="1"
 +
|-
 +
! 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==
 
==Ordered symbols==
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! Context
 
! Context
 
! Details
 
! Details
 +
! Mark
 
|-
 
|-
 
| <math>\|\cdot\|</math>  
 
| <math>\|\cdot\|</math>  
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* Real Analysis
 
* Real Analysis
 
| Denotes the [[Norm]] of a vector
 
| Denotes the [[Norm]] of a vector
 +
|
 
|-
 
|-
 
| <math>\|f\|_{C^k}</math>
 
| <math>\|f\|_{C^k}</math>
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*Functional Analysis
 
*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.
 
|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>
 
| <math>\|f\|_{L^p}</math>
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* Functional Analysis
 
* 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\|_{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>
 
| <math>\|f\|_\infty</math>
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* Real 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>
 
| 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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* Differential Geometry
 
* Differential Geometry
 
* Manifolds
 
* Manifolds
−
| That a function has continuous (partial) derivatives of all orders, it is a generalisation of <math>C^k</math> functions
+
| That a function has continuous (partial) derivatives of all orders, it is a generalisation of <math>C^k</math> functions<br/>
 +
See also [[Smooth function]] and the symbols {{M|C^\infty(\mathbb{R}^n)}} and {{M|C^\infty(M)}} where {{M|M}} is a [[Smooth manifold]]
 +
|
 +
|-
 +
| <math>C^\infty(\mathbb{R}^n)</math>
 +
|
 +
* Differential Geometry
 +
* Manifolds
 +
| The set of all [[Smooth]] functions on {{M|\mathbb{R}^n}} - see [[Smooth function]], it means {{M|f:\mathbb{R}^n\rightarrow\mathbb{R} }} 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]] {{M|M}} - see [[Smooth function]], it means {{M|f:M\rightarrow\mathbb{R} }} is smooth in the sense defined on [[Smooth function]]
 +
| TANGENT_NEW
 
|-
 
|-
 
| <math>C^k</math> ''[at {{M|p}}]''
 
| <math>C^k</math> ''[at {{M|p}}]''
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| <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/>
 
| <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]]
 
[[The set of all germs of smooth functions at a point]]
 +
|
 
|-
 
|-
 
| <math>C^k([a,b],\mathbb{R})</math>
 
| <math>C^k([a,b],\mathbb{R})</math>
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| 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/>
 
| 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
 
The unit interval will be assumed when missing
 +
|
 
|-
 
|-
 
| <math>D_a(A)</math><br/>Common: <math>D_a(\mathbb{R}^n)</math>
 
| <math>D_a(A)</math><br/>Common: <math>D_a(\mathbb{R}^n)</math>
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| 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/>
 
| 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]]
 
'''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]]
 +
| TANGENT
 
|-
 
|-
 
| <math>\mathcal{D}_a(A)</math><br/>Common: <math>\mathcal{D}_a(\mathbb{R}^n)</math>
 
| <math>\mathcal{D}_a(A)</math><br/>Common: <math>\mathcal{D}_a(\mathbb{R}^n)</math>
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* Manifolds
 
* 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)}}
 
| 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)}}
 +
| TANGENT
 
|-
 
|-
 
| <math>\bigudot_i A_i</math>
 
| <math>\bigudot_i A_i</math>
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* Measure Theory
 
* 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
 +
|
 
|-
 
|-
 
| <math>\ell^p(\mathbb{F})</math>
 
| <math>\ell^p(\mathbb{F})</math>
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*Functional Analysis
 
*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>
 
| 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>
 
| <math>\mathcal{L}^p</math>
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* Measure Theory
 
* Measure Theory
 
| Same as <math>\mathcal{L}^p</math>
 
| Same as <math>\mathcal{L}^p</math>
 +
|
 
|-
 
|-
 
| <math>T_p(A)</math><br/>Common:<math>T_p(\mathbb{R}^n)</math>
 
| <math>T_p(A)</math><br/>Common:<math>T_p(\mathbb{R}^n)</math>
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| 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 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 a germ]]
 
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 a germ]]
 +
| TANGENT
 
|}
 
|}
  

Revision as of 22:55, 12 April 2015

Ordered symbols are notations which are (likely) to appear as they are given here, for example C([a,b],R) denotes the continuous function on the interval [a,b] that map to R - this is unlikely to be given any other way because "C" is for continuous.

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 A comes before A comes before A

Expression Context Details Mark
∥⋅∥
  • Functional Analysis
  • Real Analysis
 Denotes the Norm of a vector
∥f∥Ck
  • Functional Analysis
 This Norm is defined by ∥f∥Ck=k∑i=0supt∈[0,1](|f(i)(t)|) - note f(i) is the ith derivative.
∥f∥Lp
  • Functional Analysis
 ∥f∥Lp=(∫10|f(t)|pdt)1p - it is a Norm on C([0,1],R)
∥f∥∞
  • Functional Analysis
  • Real Analysis
 It is a norm on C([a,b],R), given by ∥f∥∞=supx∈[a,b](|f(x)|)
C∞
  • Differential Geometry
  • Manifolds
 That a function has continuous (partial) derivatives of all orders, it is a generalisation of Ck functions

See also Smooth function and the symbols C∞(Rn) and C∞(M) where M is a Smooth manifold

C∞(Rn)
  • Differential Geometry
  • Manifolds
 The set of all Smooth functions on Rn - see Smooth function, it means f:Rn→R is Smooth in the usual sense - all partial derivatives of all orders are continuous. TANGENT_NEW
C∞(M)
  • Differential Geometry
  • Manifolds
 The set of all Smooth functions on the Smooth manifold M - see Smooth function, it means f:M→R is smooth in the sense defined on Smooth function TANGENT_NEW
Ck [at p]
  • Differential Geometry
  • Manifolds
 A function is said to be Ck [at p] if all (partial) derivatives of all orders exist and are continuous [at p]
C∞p
  • Differential Geometry
  • Manifolds
 C∞p(A) denotes the set of all germs of C∞ functions on A at p

The set of all germs of smooth functions at a point

Ck([a,b],R)
  • Functional Analysis
  • Real Analysis
 It is the set of all functions :[a,b]→R that are continuous and have continuous derivatives up to (and including) order k

The unit interval will be assumed when missing

Da(A)
Common: Da(Rn)
  • 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 Dp(A)

Note: This is my/Alec's notation for it, as the author[1] uses Tp(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

TANGENT
Da(A)
Common: Da(Rn)
  • 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 Tp(A) TANGENT
⋃⋅iAi
  • Measure Theory
 Makes it explicit that the items in the union (the Ai) are pairwise disjoint, that is for any two their intersection is empty
ℓp(F)
  • Functional Analysis
 The set of all bounded sequences, that is ℓp(F)={(x1,x2,...)|xi∈F, ∞∑i=1|xi|p<∞}
Lp
  • Measure Theory
 Lp(μ)={u:X→R|u∈M, ∫|u|pdμ<∞}, p∈[1,∞)⊂R

(X,A,μ) is a measure space. The class of all measurable functions for which |f|p is integrable

Lp
  • Measure Theory
 Same as Lp
Tp(A)
Common:Tp(Rn)
  • Differential Geometry
  • Manifolds
 The tangent space at a point a

Sometimes denoted Rna - Note: sometimes can mean Set of all derivations at a point which is denoted Da(Rn) and not to be confused with Da(Rn) which denotes Set of all derivations of a germ

TANGENT

Unordered symbols

Expression Context Details
A/B-measurable
  • Measure Theory
 There exists a Measurable map between the σ-algebras
a⋅b
  • Anything with vectors
 Vector dot product
  1. Jump up ↑ John M Lee - Introduction to smooth manifolds - Second edition
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