Difference between revisions of "Index of notation"

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| Properties
 
| Properties
 
| Indexed by adjectives
 
| Indexed by adjectives
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|-
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! [[Index of spaces]]
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| {{M|\mathbb{S}^n}}, {{M|l_2}}, {{M|\mathcal{C}[a,b]}}
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| Spaces
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| Index by letters
 
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Revision as of 15:10, 12 July 2015

[math]\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}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\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.

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 [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
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

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
  • Everywhere
 Denotes the set of Real numbers
[ilmath]\mathbb{S}^n[/ilmath] S_bb_N
  • Everywhere
 [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]
  • Differential Geometry
  • Manifolds
 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]
  • 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
[math]C^\infty(M)[/math]
  • 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
[math]C^k[/math] [at [ilmath]p[/ilmath]]
  • Differential Geometry
  • Manifolds
 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]
  • Differential Geometry
  • Manifolds
 [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]

The set of all germs of smooth functions at a point

[math]C^k([a,b],\mathbb{R})[/math]
  • Functional Analysis
  • Real Analysis
 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]
  • 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[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]
  • 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
[math]\bigudot_i A_i[/math]
  • Measure Theory
 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]
  • Differential Geometry
  • Manifolds
 The geometric tangent space - see Geometric Tangent Space TANGENT_NEW
[math]\ell^p(\mathbb{F})[/math]
  • Functional Analysis
 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]
  • Measure Theory
 [math]\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}[/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]
  • 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.

See The vector space of all maps between vector spaces

[math]\mathcal{L}(V)[/math]
  • Linear algebra
 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]
  • Measure Theory
 Same as [math]\mathcal{L}^p[/math]
[math]T_p(A)[/math]
Common:[math]T_p(\mathbb{R}^n)[/math]
  • 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 [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
  • Measure Theory
 There exists a Measurable map between the [ilmath]\sigma[/ilmath]-algebras
[ilmath]a\cdot b[/ilmath]
  • Anything with vectors
 Vector dot product
[math]p_0\simeq p_1\text{ rel}\{0,1\}[/math]
  • Topology
 See Homotopic paths
  1. ↑ John M Lee - Introduction to smooth manifolds - Second edition
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