Index of notation
[math]\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern13mu\cdot\mkern5mu}} }[/math][math]\newcommand{\udot}{\cup\mkern12.5mu\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.
Subindices
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 setlike notations  Things like [ilmath]\{u\le v\} [/ilmath]  setlike 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 sigmaalgebra of the real line, sometimes denoted [ilmath]\mathcal{B}(\mathbb{R})[/ilmath]. [ilmath]\mathcal{B}(X)[/ilmath] denotes the Borel sigmaalgebra generated by a topology (on) [ilmath]X[/ilmath].  [ilmath]\mathcal{B}(\cdot)[/ilmath] 
[ilmath]\mathcal{B}(\cdot)[/ilmath]  current  Denotes the Borel sigmaalgebra 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 

[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]G_p(\mathbb{R}^n)[/math] 

The geometric tangent space  see Geometric Tangent Space  TANGENT_NEW 
[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},\ \intu^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] 

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. 

[math]\mathcal{L}(V)[/math] 

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] 

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 
[math]p_0\simeq p_1\text{ rel}\{0,1\}[/math] 

See Homotopic paths 
 ↑ John M Lee  Introduction to smooth manifolds  Second edition