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 |
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 |
|
Denotes the set of Real numbers |
\mathbb{S}^n | S_bb_N |
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\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 |
|
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) |
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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) |
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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] |
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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 |
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C^\infty_p(A) denotes the set of all germs of C^\infty functions on A at p |
|
C^k([a,b],\mathbb{R}) |
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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) |
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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) |
|
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 |
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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) |
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The geometric tangent space - see Geometric Tangent Space | TANGENT_NEW |
\ell^p(\mathbb{F}) |
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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 |
|
\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) |
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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. |
|
\mathcal{L}(V) |
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Short hand for \mathcal{L}(V,V) (see above). In addition to being a vector space it is also an Algebra |
|
L^p |
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Same as \mathcal{L}^p | |
T_p(A) Common:T_p(\mathbb{R}^n) |
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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 |
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There exists a Measurable map between the \sigma-algebras |
a\cdot b |
|
Vector dot product |
p_0\simeq p_1\text{ rel}\{0,1\} |
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See Homotopic paths |
- Jump up ↑ John M Lee - Introduction to smooth manifolds - Second edition