# Index of notation

$\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}} }$$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$\require{AMScd}\newcommand{\d}[]{\mathrm{d}^{#1} }$Ordered symbols are notations which are (likely) to appear as they are given here, for example $C([a,b],\mathbb{R})$ 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 $\Vert\cdot\Vert$ Norm Not to be confused with $\vert\cdot\vert$-like expressions, see below or this index
[ilmath]\vert\cdot\vert[/ilmath] index Something like $\vert\cdot\vert$ Absolute value Not to be confused with $\Vert\cdot\Vert$-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

[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

[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

[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

[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

[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.

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

[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

[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 $\mathbb{R}$), R_bb_N is the index for $\mathbb{R}^n$

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
$\mathbb{S}^n\subset\mathbb{R}^{n+1}$ 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 $A$ comes before $\mathbb{A}$ comes before $\mathcal{A}$

Expression Context Details Mark
$C^\infty$
• Differential Geometry
• Manifolds
That a function has continuous (partial) derivatives of all orders, it is a generalisation of $C^k$ 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

$C^\infty(\mathbb{R}^n)$
• 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
$C^\infty(M)$
• 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
$C^k$ [at [ilmath]p[/ilmath]]
• Differential Geometry
• Manifolds
A function is said to be $C^k$ [at [ilmath]p[/ilmath]] if all (partial) derivatives of all orders exist and are continuous [at [ilmath]p[/ilmath]]
$C^\infty_p$
• Differential Geometry
• Manifolds
$C^\infty_p(A)$ denotes the set of all germs of $C^\infty$ functions on [ilmath]A[/ilmath] at [ilmath]p[/ilmath]
$C^k([a,b],\mathbb{R})$
• Functional Analysis
• Real Analysis
It is the set of all functions $:[a,b]\rightarrow\mathbb{R}$ that are continuous and have continuous derivatives up to (and including) order $k$

The unit interval will be assumed when missing

$D_a(A)$
Common: $D_a(\mathbb{R}^n)$
• 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
$\mathcal{D}_a(A)$
Common: $\mathcal{D}_a(\mathbb{R}^n)$
• 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
$\bigudot_i A_i$
• Measure Theory
Makes it explicit that the items in the union (the $A_i$) are pairwise disjoint, that is for any two their intersection is empty
$G_p(\mathbb{R}^n)$
• Differential Geometry
• Manifolds
The geometric tangent space - see Geometric Tangent Space TANGENT_NEW
$\ell^p(\mathbb{F})$
• Functional Analysis
The set of all bounded sequences, that is $\ell^p(\mathbb{F})=\{(x_1,x_2,...)|x_i\in\mathbb{F},\ \sum^\infty_{i=1}|x_i|^p<\infty\}$
$\mathcal{L}^p$
• Measure Theory
$\mathcal{L}^p(\mu)=\{u:X\rightarrow\mathbb{R}|u\in\mathcal{M},\ \int|u|^pd\mu<\infty\},\ p\in[1,\infty)\subset\mathbb{R}$

$(X,\mathcal{A},\mu)$ is a measure space. The class of all measurable functions for which $|f|^p$ is integrable

$\mathcal{L}(V,W)$
• 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.
$\mathcal{L}(V)$
• Linear algebra
Short hand for $\mathcal{L}(V,V)$ (see above).

In addition to being a vector space it is also an Algebra

$L^p$
• Measure Theory
Same as $\mathcal{L}^p$
$T_p(A)$
Common:$T_p(\mathbb{R}^n)$
• 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 $\mathcal{D}_a(\mathbb{R}^n)$ which denotes Set of all derivations of a germ

TANGENT

## Unordered symbols

Expression Context Details
$\mathcal{A}/\mathcal{B}$-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
$p_0\simeq p_1\text{ rel}\{0,1\}$
• Topology
See Homotopic paths
1. John M Lee - Introduction to smooth manifolds - Second edition