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== | ||
| Line 9: | Line 23: | ||
! Context | ! Context | ||
! Details | ! Details | ||
| + | ! Mark | ||
|- | |- | ||
| <math>\|\cdot\|</math> | | <math>\|\cdot\|</math> | ||
| Line 15: | Line 30: | ||
* 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> | ||
| Line 20: | Line 36: | ||
*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> | ||
| Line 25: | Line 42: | ||
* 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> | ||
| Line 31: | Line 49: | ||
* 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> | ||
| Line 36: | Line 55: | ||
* 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}}]'' | ||
| Line 50: | Line 85: | ||
| <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> | ||
| Line 57: | Line 93: | ||
| 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> | ||
| Line 64: | Line 101: | ||
| 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> | ||
| Line 70: | Line 108: | ||
* 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> | ||
| Line 75: | Line 114: | ||
* 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> | ||
| Line 80: | Line 120: | ||
*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> | ||
| Line 91: | Line 132: | ||
* 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> | ||
| Line 98: | Line 140: | ||
| 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
[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.
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]\|\cdot\|[/math] |
|
Denotes the Norm of a vector | |
| [math]\|f\|_{C^k}[/math] |
|
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}=\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] |
|
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] |
|
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] |
|
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] |
|
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]] |
|
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] |
|
[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] |
|
| [math]C^k([a,b],\mathbb{R})[/math] |
|
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] |
|
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] |
|
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] |
|
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] |
|
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(\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] |
|
Same as [math]\mathcal{L}^p[/math] | |
| [math]T_p(A)[/math] Common:[math]T_p(\mathbb{R}^n)[/math] |
|
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 |
|
There exists a Measurable map between the [ilmath]\sigma[/ilmath]-algebras |
| [ilmath]a\cdot b[/ilmath] |
|
Vector dot product |
- ↑ John M Lee - Introduction to smooth manifolds - Second edition
