Difference between revisions of "Index of notation"
m |
m |
||
| Line 1: | Line 1: | ||
| − | 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 {{M|[a,b]}} that map to {{M|\mathbb{R} }} - this is unlikely to be given any other way because "C" is for continuous. | + | {{Extra Maths}}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 {{M|[a,b]}} that map to {{M|\mathbb{R} }} - this is unlikely to be given any other way because "C" is for continuous. |
==Ordered symbols== | ==Ordered symbols== | ||
| Line 38: | Line 38: | ||
| It is the set of all functions <math>:[a,b]\rightarrow\mathbb{R}</math> that are [[Continuous map|continuous]] and have continuous derivatives up to (and including) order <math>k</math><br/> | | It is the set of all functions <math>:[a,b]\rightarrow\mathbb{R}</math> that are [[Continuous map|continuous]] and have continuous derivatives up to (and including) order <math>k</math><br/> | ||
The unit interval will be assumed when missing | The unit interval will be assumed when missing | ||
| + | |- | ||
| + | | <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>\ell^p(\mathbb{F})</math> | | <math>\ell^p(\mathbb{F})</math> | ||
Revision as of 22:35, 13 March 2015
\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.
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 |
|---|---|---|
| \|\cdot\| |
|
Denotes the Norm of a vector |
| \|f\|_{C^k} |
|
This Norm is defined by \|f\|_{C^k}=\sum^k_{i=0}\sup_{t\in[0,1]}(|f^{(i)}(t)|) - note f^{(i)} is the i^\text{th} derivative. |
| \|f\|_{L^p} |
|
\|f\|_{L^p}=\left(\int^1_0|f(t)|^pdt\right)^\frac{1}{p} - it is a Norm on \mathcal{C}([0,1],\mathbb{R}) |
| C([a,b],\mathbb{R}) |
|
It is the set of all functions :[a,b]\rightarrow\mathbb{R} that are continuous |
| C^k([a,b],\mathbb{R}) |
|
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 |
| \bigudot_i A_i | Makes it explicit that the items in the union (the A_i) are pairwise disjoint, that is for any two their intersection is empty | |
| \ell^p(\mathbb{F}) |
|
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 |
| L^p |
|
Same as \mathcal{L}^p |
Unordered symbols
| Expression | Context | Details |
|---|---|---|
| \mathcal{A}/\mathcal{B}-measurable |
|
There exists a Measurable map between the \sigma-algebras |
