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. | ||
− | == | + | ==Sub-indices== |
− | + | Due to the frequency of some things (like for example ''norms'') they have been moved to their own index. | |
+ | {| class="wikitable" border="1" | ||
+ | ! colspan="4" | Symbols | ||
+ | |- | ||
+ | ! Index | ||
+ | ! Expressions | ||
+ | ! Name | ||
+ | ! Notes | ||
+ | |- | ||
+ | ! [[Index of norms and absolute values#Norms|{{M|\Vert\cdot\Vert}} index]] | ||
+ | | Something like <math>\Vert\cdot\Vert</math> | ||
+ | | [[Norm]] | ||
+ | | Not to be confused with <math>\vert\cdot\vert</math>-like expressions, see below or [[Index of norms and absolute values#Absolute values|this index]] | ||
+ | |- | ||
+ | ! [[Index of norms and absolute values#Absolute values|{{M|\vert\cdot\vert}} 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 [[Index of norms and absolute values#Norms|this index]] | ||
+ | |- | ||
+ | ! [[Index of set-like notations]] | ||
+ | | Things like {{M|\{u\le v\} }} | ||
+ | | set-like notations | ||
+ | | WORK IN PROGRESS | ||
+ | |- | ||
+ | ! colspan="4" | Alphabetical | ||
+ | |- | ||
+ | ! Index | ||
+ | ! Expressions | ||
+ | ! Name | ||
+ | ! Notes | ||
+ | |- | ||
+ | ! [[Index of abbreviations]] | ||
+ | | WRT, AE, WTP | ||
+ | | Abbreviations | ||
+ | | Dots and case are ignored, so "wrt"="W.R.T" | ||
+ | |- | ||
+ | ! [[Index of properties]] | ||
+ | | "Closed under", "Open in" | ||
+ | | Properties | ||
+ | | Indexed by adjectives | ||
+ | |- | ||
+ | ! [[Index of spaces]] | ||
+ | | {{M|\mathbb{S}^n}}, {{M|l_2}}, {{M|\mathcal{C}[a,b]}} | ||
+ | | Spaces | ||
+ | | Index by letters | ||
+ | |} | ||
+ | ==Index== | ||
+ | Notation status meanings: | ||
+ | # ''current'' | ||
+ | #* This notation is currently used (as opposed to say archaic) unambiguous and recommended, very common | ||
+ | # ''recommended'' | ||
+ | #* This notation is recommended (which means it is also currently used (otherwise it'd simply be: suggested)) as other notations for the same thing have problems (such as ambiguity) | ||
+ | # ''suggested'' | ||
+ | #* This notation is clear (in line with the [[Doctrine of least surprise]]) and will cause no problems but is uncommon | ||
+ | # ''archaic'' | ||
+ | #* This is an old notation for something and no longer used (or rarely used) in current mathematics | ||
+ | # ''dangerous'' | ||
+ | #* This notation is ambiguous, or likely to cause problems when read by different people and therefore should not be used. | ||
+ | ===Notations starting with B=== | ||
+ | {{:Index of notation/B}} | ||
+ | ===Notations starting with C=== | ||
+ | {{:Index of notation/C}} | ||
+ | ===Notations starting with L=== | ||
+ | {{:Index of notation/L}} | ||
+ | ===Notations starting with N=== | ||
+ | {{:Index of notation/N}} | ||
+ | ===Notations starting with P=== | ||
+ | {{:Index of notation/P}} | ||
+ | ===Notations starting with Q=== | ||
+ | {{:Index of notation/Q}} | ||
+ | ===Notations starting with R=== | ||
+ | {{:Index of notation/R}} | ||
+ | ===Old stuff=== | ||
+ | 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" | {| class="wikitable" border="1" | ||
|- | |- | ||
! Expression | ! Expression | ||
+ | ! Index | ||
! Context | ! Context | ||
! Details | ! Details | ||
|- | |- | ||
− | | | + | | {{M|\mathbb{R} }} |
+ | | R_bb | ||
| | | | ||
− | * | + | * Everywhere |
− | + | | Denotes the set of [[Real numbers]] | |
− | | Denotes the [[ | + | |
|- | |- | ||
− | | | + | | {{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== | ||
+ | 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== | ||
+ | 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 | ||
+ | ! Mark | ||
|- | |- | ||
| <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>G_p(\mathbb{R}^n)</math> | ||
+ | | | ||
+ | * Differential Geometry | ||
+ | * Manifolds | ||
+ | | The geometric tangent space - see [[Tangent space#Geometric Tangent Space|Geometric Tangent Space]] | ||
+ | | TANGENT_NEW | ||
|- | |- | ||
| <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> | ||
Line 86: | Line 207: | ||
| <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>\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>(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>\mathcal{L}(V,W)</math> | ||
+ | | | ||
+ | * Linear Algebra | ||
+ | | The set of all [[Linear map|linear maps]] from a [[Vector space|vector space]] {{M|V}} (over a [[Field|field]] {{M|F}}) and another vector space {{M|W}} also over {{M|F}}. It is a vector space itself.<br/> | ||
+ | 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).<br/> | ||
+ | In addition to being a vector space it is also an [[Algebra]] | ||
+ | | | ||
|- | |- | ||
| <math>L^p</math> | | <math>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 | ||
|} | |} | ||
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* Anything with vectors | * Anything with vectors | ||
| [[Vector dot product]] | | [[Vector dot product]] | ||
+ | |- | ||
+ | | <math>p_0\simeq p_1\text{ rel}\{0,1\}</math> | ||
+ | | | ||
+ | * Topology | ||
+ | | See [[Homotopic paths]] | ||
|} | |} | ||
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[[Category:Subjects]] | [[Category:Subjects]] | ||
+ | [[Category:Index]] |
Latest revision as of 06:13, 1 January 2017
[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 |
Index of set-like notations | Things like [ilmath]\{u\le v\} [/ilmath] | set-like notations | WORK IN PROGRESS |
Alphabetical | |||
Index | Expressions | Name | Notes |
Index of abbreviations | WRT, AE, WTP | Abbreviations | Dots and case are ignored, so "wrt"="W.R.T" |
Index of properties | "Closed under", "Open in" | Properties | Indexed by adjectives |
Index of spaces | [ilmath]\mathbb{S}^n[/ilmath], [ilmath]l_2[/ilmath], [ilmath]\mathcal{C}[a,b][/ilmath] | Spaces | Index by letters |
Index
Notation status meanings:
- current
- This notation is currently used (as opposed to say archaic) unambiguous and recommended, very common
- recommended
- This notation is recommended (which means it is also currently used (otherwise it'd simply be: suggested)) as other notations for the same thing have problems (such as ambiguity)
- suggested
- This notation is clear (in line with the Doctrine of least surprise) and will cause no problems but is uncommon
- archaic
- This is an old notation for something and no longer used (or rarely used) in current mathematics
- dangerous
- This notation is ambiguous, or likely to cause problems when read by different people and therefore should not be used.
Notations starting with B
Expression | Status | Meanings | See also |
---|---|---|---|
[ilmath]\mathcal{B} [/ilmath] | current | The Borel sigma-algebra of the real line, sometimes denoted [ilmath]\mathcal{B}(\mathbb{R})[/ilmath]. [ilmath]\mathcal{B}(X)[/ilmath] denotes the Borel sigma-algebra generated by a topology (on) [ilmath]X[/ilmath]. | [ilmath]\mathcal{B}(\cdot)[/ilmath] |
[ilmath]\mathcal{B}(\cdot)[/ilmath] | current | Denotes the Borel sigma-algebra generated by [ilmath]\cdot[/ilmath]. Here the "[ilmath]\cdot[/ilmath]" is any topological space, for a topology [ilmath](X,\mathcal{J})[/ilmath] we usually still write [ilmath]\mathcal{B}(X)[/ilmath] however if dealing with multiple topologies on [ilmath]X[/ilmath] writing [ilmath]\mathcal{B}(\mathcal{J})[/ilmath] is okay. If the topology is the real line with the usual (euclidean) topology, we simply write [ilmath]\mathcal{B} [/ilmath] | [ilmath]\mathcal{B} [/ilmath] |
Notations starting with C
Expression | Status | Meanings | See also |
---|---|---|---|
[ilmath]C(X,Y)[/ilmath] | current | The set of continuous functions between topological spaces. There are many special cases of what [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] might be, for example: [ilmath]C(I,X)[/ilmath] - all paths in [ilmath](X,\mathcal{ J })[/ilmath]. These sets often have additional structure (eg, vector space, algebra)
Index of notation for sets of continuous maps:
|
Notations starting with L
Expression | Status | Meanings | See also | |
---|---|---|---|---|
[ilmath]L[/ilmath] (Linear Algebra) |
[ilmath]L(V,W)[/ilmath] | current | Set of all linear maps, [ilmath](:V\rightarrow W)[/ilmath] - is a vector space in own right. Both vec spaces need to be over the same field, say [ilmath]\mathbb{F} [/ilmath]. | |
[ilmath]L(V)[/ilmath] | current | Shorthand for [ilmath]L(V,V)[/ilmath] - see above | ||
[ilmath]L(V,\mathbb{F})[/ilmath] | current | Space of all linear functionals, ie linear maps of the form [ilmath](:V\rightarrow\mathbb{F})[/ilmath] as every field is a vector space, this is no different to [ilmath]L(V,W)[/ilmath].
|
||
[ilmath]L(V_1,\ldots,V_k;W)[/ilmath] | current | All multilinear maps of the form [ilmath](:V_1\times\cdots\times V_k\rightarrow W)[/ilmath] | ||
[ilmath]L(V_1,\ldots,V_k;\mathbb{F})[/ilmath] | current | Special case of [ilmath]L(V_1,\ldots,V_k;W)[/ilmath] as every field is a vector space. Has relations to the tensor product | ||
[ilmath]\mathcal{L}(\cdots)[/ilmath] | current | Same as version above, with requirement that the maps be continuous, requires the vector spaces to be normed spaces (which is where the metric comes from to yield a topology for continuity to make sense) | ||
[ilmath]L[/ilmath] (Measure Theory / Functional Analysis) |
[ilmath]L^p[/ilmath] | current | TODO: todo
|
|
[ilmath]\ell^p[/ilmath] | current | Special case of [ilmath]L^p[/ilmath] on [ilmath]\mathbb{N} [/ilmath] |
Notations starting with N
Expression | Status | Meanings | See also |
---|---|---|---|
[ilmath]\mathbb{N} [/ilmath] | current | The natural number (or naturals), either [ilmath]\mathbb{N}:=\{0,1,\ldots,n,\ldots\}[/ilmath] or [ilmath]\mathbb{N}:=\{1,2,\ldots,n,\ldots\}[/ilmath]. In contexts where starting from one actually matters [ilmath]\mathbb{N}_+[/ilmath] is used, usually it is clear from the context, [ilmath]\mathbb{N}_0[/ilmath] may be used when the 0 being present is important. |
|
[ilmath]\mathbb{N}_+[/ilmath] | current | Used if it is important to consider the naturals as the set [ilmath]\{1,2,\ldots\} [/ilmath], it's also an example of why the notation [ilmath]\mathbb{R}_+[/ilmath] is bad (as some authors use [ilmath]\mathbb{R}_+:=\{x\in\mathbb{R}\ \vert\ x\ge 0\}[/ilmath] here it is being used for [ilmath]>0[/ilmath]) |
|
[ilmath]\mathbb{N}_0[/ilmath] | current | Used if it is important to consider the naturals as the set [ilmath]\{0,1,\ldots\} [/ilmath] |
|
Notations starting with P
Expression | Status | Meanings | See also |
---|---|---|---|
[ilmath]p[/ilmath] | current | Prime numbers, projective functions (along with [ilmath]\pi[/ilmath]), vector points (typically [ilmath]p,q,r[/ilmath]), representing rational numbers as [ilmath]\frac{p}{q} [/ilmath] | |
[ilmath]P[/ilmath] | dangerous | Sometimes used for probability measures, the notation [ilmath]\mathbb{P} [/ilmath] is recommended for these. | |
[ilmath]\mathbb{P} [/ilmath] | current | See P (notation) for more information. Typically:
TODO: Introduction to Lattices and Order - p2 for details, bottom of page
TODO: Find refs |
|
[ilmath]\mathcal{P}(X)[/ilmath] | current | Power set, I have seen no other meaning for [ilmath]\mathcal{P}(X)[/ilmath] (where [ilmath]X[/ilmath] is a set) however I have seen the notation:
|
Notations starting with Q
Expression | Status | Meanings | See also |
---|---|---|---|
[ilmath]\mathbb{Q} [/ilmath] | current | The quotient field, the field of rational numbers, or simply the rationals. A subset of the reals ([ilmath]\mathbb{R} [/ilmath]) |
Notations starting with R
Expression | Status | Meanings | See also |
---|---|---|---|
[ilmath]\mathbb{R} [/ilmath] | current | Real numbers | |
[ilmath]\mathbb{R}_+[/ilmath] | dangerous | See [ilmath]\mathbb{R}_+[/ilmath] (notation) for details on why this is bad. It's a very ambiguous notation, use [ilmath]\mathbb{R}_{\ge 0} [/ilmath] or [ilmath]\mathbb{R}_{>0} [/ilmath] instead. |
|
[ilmath]\mathbb{R}_{\ge 0} [/ilmath] | recommended | [ilmath]:=\{x\in\mathbb{R}\ \vert\ x\ge 0\}[/ilmath], recommended over the dangerous notation of [ilmath]\mathbb{R}_+[/ilmath], see details there. |
|
[ilmath]\mathbb{R}_{>0} [/ilmath] | recommended | [ilmath]:=\{x\in\mathbb{R}\ \vert\ x>0[/ilmath], recommended over the dangerous notation of [ilmath]\mathbb{R}_+[/ilmath], see details there. |
|
[ilmath]\mathbb{R}_{\le x},\ \mathbb{R}_{\ge x} [/ilmath], so forth | recommended | Recommended notations for rays of the real line. See Denoting commonly used subsets of [ilmath]\mathbb{R} [/ilmath] |
|
Old stuff
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 |
|
Denotes the set of Real numbers |
[ilmath]\mathbb{S}^n[/ilmath] | S_bb_N |
|
[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 |
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[math]C^\infty[/math] |
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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 |
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[math]C^\infty(\mathbb{R}^n)[/math] |
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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] |
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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]] |
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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] |
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[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] |
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[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 and have continuous derivatives up to (and including) order [math]k[/math] The unit interval will be assumed when missing |
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[math]D_a(A)[/math] 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 [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] |
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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] |
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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] |
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The geometric tangent space - see Geometric Tangent Space | TANGENT_NEW |
[math]\ell^p(\mathbb{F})[/math] |
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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] |
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[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 |
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[math]\mathcal{L}(V,W)[/math] |
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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. |
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[math]\mathcal{L}(V)[/math] |
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Short hand for [math]\mathcal{L}(V,V)[/math] (see above). In addition to being a vector space it is also an Algebra |
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[math]L^p[/math] |
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Same as [math]\mathcal{L}^p[/math] | |
[math]T_p(A)[/math] Common:[math]T_p(\mathbb{R}^n)[/math] |
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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 |
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There exists a Measurable map between the [ilmath]\sigma[/ilmath]-algebras |
[ilmath]a\cdot b[/ilmath] |
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Vector dot product |
[math]p_0\simeq p_1\text{ rel}\{0,1\}[/math] |
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See Homotopic paths |
- ↑ John M Lee - Introduction to smooth manifolds - Second edition