Index of notation

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\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
Alphabetical
Index Expressions Name Notes
Index of abbreviations WRT, AE, WTP Abbreviations Dots and case are ignored, so "wrt"="W.R.T"

Index

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.

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 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[1] 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
  1. Jump up John M Lee - Introduction to smooth manifolds - Second edition