Difference between revisions of "Index of notation"

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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]]
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|-
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! [[Index of set-like notations]]
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| Things like {{M|\{u\le v\} }}
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| set-like notations
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| WORK IN PROGRESS
 
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! colspan="4" | Alphabetical
 
! colspan="4" | Alphabetical
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# ''dangerous''
 
# ''dangerous''
 
#* This notation is ambiguous, or likely to cause problems when read by different people and therefore should not be used.
 
#* This notation is ambiguous, or likely to cause problems when read by different people and therefore should not be used.
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===Notations starting with B===
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{{:Index of notation/B}}
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===Notations starting with C===
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{{:Index of notation/C}}
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===Notations starting with L===
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{{:Index of notation/L}}
 
===Notations starting with N===
 
===Notations starting with N===
 
{{:Index of notation/N}}
 
{{:Index of notation/N}}
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===Notations starting with P===
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===Notations starting with Q===
 
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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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