Difference between revisions of "Index of notation"
m |
m |
||
| Line 134: | Line 134: | ||
| <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><br/> | | <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><br/> | ||
<math>(X,\mathcal{A},\mu)</math> is a [[Measure space|measure space]]. The class of all [[Measurable function|measurable functions]] for which <math>|f|^p</math> is integrable | <math>(X,\mathcal{A},\mu)</math> is a [[Measure space|measure space]]. The class of all [[Measurable function|measurable functions]] for which <math>|f|^p</math> is integrable | ||
| + | | | ||
| + | |- | ||
| + | | <math>\mathcal{L}(V,W)</math> | ||
| + | | | ||
| + | * Linear Algebra | ||
| + | | The set of all [[Linear map|linear maps]] from a [[Vector space|vector space]] {{M|V}} (over a [[Field|field]] {{M|F}}) and another vector space {{M|W}} also over {{M|F}}. It is a vector space itself.<br/> | ||
| + | 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).<br/> | ||
| + | In addition to being a vector space it is also an [[Algebra]] | ||
| + | | | ||
|- | |- | ||
| <math>L^p</math> | | <math>L^p</math> | ||
Revision as of 18:25, 10 May 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.
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]\|\cdot\|[/math] |
|
Denotes the Norm of a vector | |
| [math]\|f\|_{C^k}[/math] |
|
This Norm is defined by [math]\|f\|_{C^k}=\sum^k_{i=0}\sup_{t\in[0,1]}(|f^{(i)}(t)|)[/math] - note [math]f^{(i)}[/math] is the [math]i^\text{th}[/math] derivative. | |
| [math]\|f\|_{L^p}[/math] |
|
[math]\|f\|_{L^p}=\left(\int^1_0|f(t)|^pdt\right)^\frac{1}{p}[/math] - it is a Norm on [math]\mathcal{C}([0,1],\mathbb{R})[/math] | |
| [math]\|f\|_\infty[/math] |
|
It is a norm on [math]C([a,b],\mathbb{R})[/math], given by [math]\|f\|_\infty=\sup_{x\in[a,b]}(|f(x)|)[/math] | |
| [math]C^\infty[/math] |
|
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] |
|
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] |
|
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]] |
|
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] |
|
[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] |
|
| [math]C^k([a,b],\mathbb{R})[/math] |
|
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] |
|
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] |
|
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] |
|
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] |
|
The geometric tangent space - see Geometric Tangent Space | TANGENT_NEW |
| [math]\ell^p(\mathbb{F})[/math] |
|
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] |
|
[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] |
|
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. |
|
| [math]\mathcal{L}(V)[/math] |
|
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] |
|
Same as [math]\mathcal{L}^p[/math] | |
| [math]\mathbb{S}^n[/math] |
|
[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. |
|
| [math]T_p(A)[/math] Common:[math]T_p(\mathbb{R}^n)[/math] |
|
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 |
|
There exists a Measurable map between the [ilmath]\sigma[/ilmath]-algebras |
| [ilmath]a\cdot b[/ilmath] |
|
Vector dot product |
| [math]p_0\simeq p_1\text{ rel}\{0,1\}[/math] |
|
See Homotopic paths |
- ↑ John M Lee - Introduction to smooth manifolds - Second edition
