Index of notation

[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
   • This notation is clear (in line with the Doctrine of least surprise) and will cause no problems but is uncommon
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: [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. [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] [ilmath]C(X,\mathbb{R})[/ilmath] - The algebra of all real functionals on [ilmath]X[/ilmath]. [ilmath]\mathbb{R} [/ilmath] considered with usual topology See also: [ilmath]C(X,\mathbb{K})[/ilmath] [ilmath]C(X,\mathbb{C})[/ilmath] - The algebra of all complex functionals on [ilmath]X[/ilmath]. [ilmath]\mathbb{C} [/ilmath] considered with usual topology See also: [ilmath]C(X,\mathbb{K})[/ilmath] [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. [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. [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]. [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]. [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]. [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]. See - Dual vector space, written [ilmath]V^*[/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: [ilmath]\mathbb{P} [/ilmath] denotes a probability function almost always. However I have seen: [ilmath]\mathbb{P} [/ilmath] to denote set of predicates in order theory TODO: Introduction to Lattices and Order - p2 for details, bottom of page Also used in denoting the real projective spaces, eg [ilmath]\mathbb{RP}^2[/ilmath] for the real projective plane ([ilmath]P\mathbb{R}^2[/ilmath] is also used) 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.

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 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