Index of notation

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$ 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 $[a,b]$ that map to $\mathbb{R}$ - 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
$\Vert\cdot\Vert$ index	Something like $$\Vert\cdot\Vert$$	Norm	Not to be confused with $$\vert\cdot\vert$$ -like expressions, see below or this index
$\vert\cdot\vert$ 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 $\{u\le v\}$	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	$\mathbb{S}^n$, $l_2$, $\mathcal{C}[a,b]$	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
$\mathcal{B}$	current	The Borel sigma-algebra of the real line, sometimes denoted $\mathcal{B}(\mathbb{R})$. $\mathcal{B}(X)$ denotes the Borel sigma-algebra generated by a topology (on) $X$.	$\mathcal{B}(\cdot)$
$\mathcal{B}(\cdot)$	current	Denotes the Borel sigma-algebra generated by $\cdot$. Here the "$\cdot$" is any topological space, for a topology $(X,\mathcal{J})$ we usually still write $\mathcal{B}(X)$ however if dealing with multiple topologies on $X$ writing $\mathcal{B}(\mathcal{J})$ is okay. If the topology is the real line with the usual (euclidean) topology, we simply write $\mathcal{B}$	$\mathcal{B}$

Notations starting with C

Expression	Status	Meanings	See also
$C(X,Y)$	current	The set of continuous functions between topological spaces. There are many special cases of what $X$ and $Y$ might be, for example: $C(I,X)$ - all paths in $(X,\mathcal{ J })$. 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: [Expand]Index of notation for sets of continuous maps: $C(X,Y)$ - for topological spaces $(X,\mathcal{ J })$ and $(Y,\mathcal{ K })$, $C(X,Y)$ is the set of all continuous maps between them. $C(I,X)$ - $I:\eq[0,1]\subset\mathbb{R}$, set of all paths on a topological space $(X,\mathcal{ J })$ Sometimes written: $C([0,1],X)$ $C(X,\mathbb{R})$ - The algebra of all real functionals on $X$. $\mathbb{R}$ considered with usual topology See also: $C(X,\mathbb{K})$ $C(X,\mathbb{C})$ - The algebra of all complex functionals on $X$. $\mathbb{C}$ considered with usual topology See also: $C(X,\mathbb{K})$ $C(X,\mathbb{K})$ - The algebra of all functionals on $X$, where $\mathbb{K}$ is either the reals, $\mathbb{R}$ or the complex numbers, $\mathbb{C}$, equipped with their usual topology. $C(X,\mathbb{F})$ - structure unsure at time of writing - set of all continuous functions of the form $f:X\rightarrow\mathbb{F}$ where $\mathbb{F}$ is any field with an absolute value, with the topology that absolute value induces. $C(K,\mathbb{R})$ - $K$ must be a compact topological space. Denotes the algebra of real functionals from $K$ to $\mathbb{R}$ - in line with the notation $C(X,\mathbb{R})$. $C(K,\mathbb{C})$ - $K$ must be a compact topological space. Denotes the algebra of complex functionals from $K$ to $\mathbb{C}$ - in line with the notation $C(X,\mathbb{C})$. $C(K,\mathbb{K})$ - $K$ must be a compact topological space. Denotes either $C(K,\mathbb{R})$ or $C(K,\mathbb{C})$ - we do not care/specify the particular field - in line with the notation $C(X,\mathbb{K})$. $C(K,\mathbb{F})$ - denotes that the space $K$ is a compact topological space, the meaning of the field corresponds to the definitions for $C(X,\mathbb{F})$ as given above for that field - in line with the notation $C(X,\mathbb{F})$.

Notations starting with L

Expression		Status	Meanings	See also
$L$ (Linear Algebra)	$L(V,W)$	current	Set of all linear maps, $(:V\rightarrow W)$ - is a vector space in own right. Both vec spaces need to be over the same field, say $\mathbb{F}$.
	$L(V)$	current	Shorthand for $L(V,V)$ - see above
	$L(V,\mathbb{F})$	current	Space of all linear functionals, ie linear maps of the form $(:V\rightarrow\mathbb{F})$ as every field is a vector space, this is no different to $L(V,W)$. See - Dual vector space, written $V^*$
	$L(V_1,\ldots,V_k;W)$	current	All multilinear maps of the form $(:V_1\times\cdots\times V_k\rightarrow W)$
	$L(V_1,\ldots,V_k;\mathbb{F})$	current	Special case of $L(V_1,\ldots,V_k;W)$ as every field is a vector space. Has relations to the tensor product
	$\mathcal{L}(\cdots)$	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)
$L$ (Measure Theory / Functional Analysis)	$L^p$	current	TODO: todo
	$\ell^p$	current	Special case of $L^p$ on $\mathbb{N}$

Notations starting with N

Expression	Status	Meanings	See also
$\mathbb{N}$	current	The natural number (or naturals), either $\mathbb{N}:=\{0,1,\ldots,n,\ldots\}$ or $\mathbb{N}:=\{1,2,\ldots,n,\ldots\}$. In contexts where starting from one actually matters $\mathbb{N}_+$ is used, usually it is clear from the context, $\mathbb{N}_0$ may be used when the 0 being present is important.	$\mathbb{N}_+$ $\mathbb{N}_0$
$\mathbb{N}_+$	current	Used if it is important to consider the naturals as the set $\{1,2,\ldots\}$, it's also an example of why the notation $\mathbb{R}_+$ is bad (as some authors use $\mathbb{R}_+:=\{x\in\mathbb{R}\ \vert\ x\ge 0\}$ here it is being used for $>0$)	$\mathbb{N}_0$
$\mathbb{N}_0$	current	Used if it is important to consider the naturals as the set $\{0,1,\ldots\}$	$\mathbb{N}_+$

Notations starting with P

Expression	Status	Meanings	See also
$p$	current	Prime numbers, projective functions (along with $\pi$), vector points (typically $p,q,r$), representing rational numbers as $\frac{p}{q}$
$P$	dangerous	Sometimes used for probability measures, the notation $\mathbb{P}$ is recommended for these.
$\mathbb{P}$	current	See P (notation) for more information. Typically: $\mathbb{P}$ denotes a probability function almost always. However I have seen: $\mathbb{P}$ 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 $\mathbb{RP}^2$ for the real projective plane ($P\mathbb{R}^2$ is also used) TODO: Find refs
$\mathcal{P}(X)$	current	Power set, I have seen no other meaning for $\mathcal{P}(X)$ (where $X$ is a set) however I have seen the notation: $2^X:=\mathcal{P}(X)$ used to denote powerset

Notations starting with Q

Expression	Status	Meanings	See also
$\mathbb{Q}$	current	The quotient field, the field of rational numbers, or simply the rationals. A subset of the reals ($\mathbb{R}$)

Notations starting with R

Expression	Status	Meanings	See also
$\mathbb{R}$	current	Real numbers
$\mathbb{R}_+$	dangerous	See $\mathbb{R}_+$ (notation) for details on why this is bad. It's a very ambiguous notation, use $\mathbb{R}_{\ge 0}$ or $\mathbb{R}_{>0}$ instead.	$\mathbb{R}_{\ge 0}$ $\mathbb{R}_{> 0}$
$\mathbb{R}_{\ge 0}$	recommended	$:=\{x\in\mathbb{R}\ \vert\ x\ge 0\}$, recommended over the dangerous notation of $\mathbb{R}_+$, see details there.	$\mathbb{R}_+$ $\mathbb{R}_{>0}$
$\mathbb{R}_{>0}$	recommended	$:=\{x\in\mathbb{R}\ \vert\ x>0$, recommended over the dangerous notation of $\mathbb{R}_+$, see details there.	$\mathbb{R}_+$ $\mathbb{R}_{\ge 0}$
$\mathbb{R}_{\le x},\ \mathbb{R}_{\ge x}$, so forth	recommended	Recommended notations for rays of the real line. See Denoting commonly used subsets of $\mathbb{R}$	$\mathbb{R}_+$

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
$\mathbb{R}$	R_bb	Everywhere	Denotes the set of Real numbers
$\mathbb{S}^n$	S_bb_N	Everywhere	$$\mathbb{S}^n\subset\mathbb{R}^{n+1}$$ and is the $n$-sphere, examples: $\mathbb{S}^1$ is a circle, $\mathbb{S}^2$ is a sphere, $\mathbb{S}^0$ 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 $C^\infty(\mathbb{R}^n)$ and $C^\infty(M)$ where $M$ is a Smooth manifold
$$C^\infty(\mathbb{R}^n)$$	Differential Geometry Manifolds	The set of all Smooth functions on $\mathbb{R}^n$ - see Smooth function, it means $f:\mathbb{R}^n\rightarrow\mathbb{R}$ 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 $M$ - see Smooth function, it means $f:M\rightarrow\mathbb{R}$ is smooth in the sense defined on Smooth function	TANGENT_NEW
$$C^k$$ [at $p$]	Differential Geometry Manifolds	A function is said to be $$C^k$$ [at $p$] if all (partial) derivatives of all orders exist and are continuous [at $p$]
$$C^\infty_p$$	Differential Geometry Manifolds	$$C^\infty_p(A)$$ denotes the set of all germs of $$C^\infty$$ functions on $A$ at $p$ The set of all germs of smooth functions at a point
$$C^k([a,b],\mathbb{R})$$	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 $\mathcal{D}_p(A)$ Note: This is my/Alec's notation for it, as the author uses $T_p(A)$ - 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 $T_p(A)$	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 $V$ (over a field $F$) and another vector space $W$ also over $F$. It is a vector space itself. See The vector space of all maps between vector spaces
$$\mathcal{L}(V)$$	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 $a$ Sometimes denoted $\mathbb{R}^n_a$ - Note: sometimes can mean Set of all derivations at a point which is denoted $D_a(\mathbb{R}^n)$ 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 $\sigma$-algebras
$a\cdot b$	Anything with vectors	Vector dot product
$$p_0\simeq p_1\text{ rel}\{0,1\}$$	Topology	See Homotopic paths