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]]
 +
|-
 +
! [[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"
 +
|-
 +
! 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.
 
 
|
 
|
 
|-
 
|-
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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:

  1. current
    • This notation is currently used (as opposed to say archaic) unambiguous and recommended, very common
  2. 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)
  3. suggested
  4. archaic
    • This is an old notation for something and no longer used (or rarely used) in current mathematics
  5. 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)


These spaces may not directly be topological spaces, they may be metric spaces, or normed spaces or inner-product spaces, these of course do have a natural topology associated with them, and it is with respect to that we refer.


- see Index of notation for sets of continuous maps. Transcluded below for convenience:
Index of notation for sets of continuous maps:
  1. [ilmath]C(X,Y)[/ilmath] - for topological spaces [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath], [ilmath]C(X,Y)[/ilmath] is the set of all continuous maps between them.
  2. [ilmath]C(I,X)[/ilmath] - [ilmath]I:\eq[0,1]\subset\mathbb{R} [/ilmath], set of all paths on a topological space [ilmath](X,\mathcal{ J })[/ilmath]
    • Sometimes written: [ilmath]C([0,1],X)[/ilmath]
  3. [ilmath]C(X,\mathbb{R})[/ilmath] - The algebra of all real functionals on [ilmath]X[/ilmath]. [ilmath]\mathbb{R} [/ilmath] considered with usual topology
  4. [ilmath]C(X,\mathbb{C})[/ilmath] - The algebra of all complex functionals on [ilmath]X[/ilmath]. [ilmath]\mathbb{C} [/ilmath] considered with usual topology
  5. [ilmath]C(X,\mathbb{K})[/ilmath] - The algebra of all functionals on [ilmath]X[/ilmath], where [ilmath]\mathbb{K} [/ilmath] is either the reals, [ilmath]\mathbb{R} [/ilmath] or the complex numbers, [ilmath]\mathbb{C} [/ilmath], equipped with their usual topology.
  6. [ilmath]C(X,\mathbb{F})[/ilmath] - structure unsure at time of writing - set of all continuous functions of the form [ilmath]f:X\rightarrow\mathbb{F} [/ilmath] where [ilmath]\mathbb{F} [/ilmath] is any field with an absolute value, with the topology that absolute value induces.
  7. [ilmath]C(K,\mathbb{R})[/ilmath] - [ilmath]K[/ilmath] must be a compact topological space. Denotes the algebra of real functionals from [ilmath]K[/ilmath] to [ilmath]\mathbb{R} [/ilmath] - in line with the notation [ilmath]C(X,\mathbb{R})[/ilmath].
  8. [ilmath]C(K,\mathbb{C})[/ilmath] - [ilmath]K[/ilmath] must be a compact topological space. Denotes the algebra of complex functionals from [ilmath]K[/ilmath] to [ilmath]\mathbb{C} [/ilmath] - in line with the notation [ilmath]C(X,\mathbb{C})[/ilmath].
  9. [ilmath]C(K,\mathbb{K})[/ilmath] - [ilmath]K[/ilmath] must be a compact topological space. Denotes either [ilmath]C(K,\mathbb{R})[/ilmath] or [ilmath]C(K,\mathbb{C})[/ilmath] - we do not care/specify the particular field - in line with the notation [ilmath]C(X,\mathbb{K})[/ilmath].
  10. [ilmath]C(K,\mathbb{F})[/ilmath] - denotes that the space [ilmath]K[/ilmath] is a compact topological space, the meaning of the field corresponds to the definitions for [ilmath]C(X,\mathbb{F})[/ilmath] as given above for that field - in line with the notation [ilmath]C(X,\mathbb{F})[/ilmath].

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]
  • [ilmath]\mathbb{N}_0[/ilmath]
[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]
[ilmath]\mathbb{N}_0[/ilmath] current Used if it is important to consider the naturals as the set [ilmath]\{0,1,\ldots\} [/ilmath]
  • [ilmath]\mathbb{N}_+[/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:
  • [ilmath]2^X:=\mathcal{P}(X)[/ilmath] used to denote powerset

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]
  • [ilmath]\mathbb{R}_{> 0} [/ilmath]
[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}_+[/ilmath]
  • [ilmath]\mathbb{R}_{>0} [/ilmath]
[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}_+[/ilmath]
  • [ilmath]\mathbb{R}_{\ge 0} [/ilmath]
[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]
  • [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
  • 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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