Difference between revisions of "Index of notation"

Revision as of 00:13, 5 April 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
$\\|\cdot\\|$	Functional Analysis Real Analysis	Denotes the Norm of a vector
$\\|f\\|_{C^k}$	Functional Analysis	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.
$\\|f\\|_{L^p}$	Functional Analysis	$\\|f\\|_{L^p}=\left(\int^1_0\|f(t)\|^pdt\right)^\frac{1}{p}$ - it is a Norm on $\mathcal{C}([0,1],\mathbb{R})$
$\\|f\\|_\infty$	Functional Analysis Real Analysis	It is a norm on $C([a,b],\mathbb{R})$ , given by $\\|f\\|_\infty=\sup_{x\in[a,b]}(\|f(x)\|)$
$C^\infty$	Differential Geometry Manifolds	That a function has continuous (partial) derivatives of all orders, it is a generalisation of $C^k$ functions
$C^k$ [at $p$ ]	Differential Geometry Manifolds	A function is said to be [at $p$ ] if all (partial) derivatives of all orders exist and are continuous [at $p$ ]
$C^k([a,b],\mathbb{R})$	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
$\mathcal{D}_a(\mathbb{R}^n)$	Differential Geometry Manifolds	Denotes Set of all derivations at a point - sometimes denoted $T_a(\mathbb{R}^n)$ (and such authors will denote the tangent space as $\mathbb{R}^n_a$ )
$\bigudot_i A_i$		Makes it explicit that the items in the union (the $A_i$ ) are pairwise disjoint, that is for any two their intersection is empty
$\ell^p(\mathbb{F})$	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	$(X,\mathcal{A},\mu)$ is a measure space. The class of all measurable functions for which $\|f\|^p$ is integrable
$L^p$	Measure Theory	Same as $\mathcal{L}^p$
$T_p(\mathbb{R}^n)$	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 often denoted $\mathcal{D}_a(\mathbb{R}^n)$

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

@@ Line 31: / Line 31: @@
 * Real Analysis
 | 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>
+|
+* Differential Geometry
+* Manifolds
+| That a function has continuous (partial) derivatives of all orders, it is a generalisation of <math>C^k</math> functions
+|-
+| <math>C^k</math> ''[at {{M|p}}]''
+|
+* Differential Geometry
+* Manifolds
+| A function is said to be <math>C^k</math> [at {{M|p}}] if all (partial) derivatives of all orders exist and are continuous [at {{M|p}}]
 |-
 | <math>C^k([a,b],\mathbb{R})</math>
@@ Line 38: / Line 50: @@
 | 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
+|-
+| <math>\mathcal{D}_a(\mathbb{R}^n)</math>
+|
+* Differential Geometry
+* Manifolds
+| Denotes [[Set of all derivations at a point]] - sometimes denoted {{M|T_a(\mathbb{R}^n)}} (and such authors will denote the tangent space as {{M|\mathbb{R}^n_a}})
 |-
 | <math>\bigudot_i A_i</math>
@@ Line 58: / Line 76: @@
 * Measure Theory
 | Same as <math>\mathcal{L}^p</math>
+|-
+| <math>T_p(\mathbb{R}^n)</math>
+|
+* Differential Geometry
+* Manifolds
+| The [[Tangent space|tangent space]] at a point {{M|a}}<br />
+Sometimes denoted {{M|\mathbb{R}^n_a}} - '''Note:''' sometimes can mean [[Set of all derivations at a point]] which is often denoted {{M|\mathcal{D}_a(\mathbb{R}^n)}}
 |}

Difference between revisions of "Index of notation"

Revision as of 00:13, 5 April 2015

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