Difference between revisions of "Index of notation"

Revision as of 02:55, 5 April 2015

$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$ Ordered symbols are notations which are (likely) to appear as they are given here, for example 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 $p$ ]
$C^\infty_p$	Differential Geometry Manifolds	denotes the set of all germs of functions on $A$ at $p$ The set of all germs of smooth functions at a point
$C^k([a,b],\mathbb{R})$	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
$D_a(A)$ Common: $D_a(\mathbb{R}^n)$	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 $\mathcal{D}_p(A)$ 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
$\mathcal{D}_a(A)$ Common: $\mathcal{D}_a(\mathbb{R}^n)$	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 $T_p(A)$
$\bigudot_i A_i$	Measure Theory	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	$\mathcal{L}^p(\mu)=\{u:X\rightarrow\mathbb{R}\|u\in\mathcal{M},\ \int\|u\|^pd\mu<\infty\},\ p\in[1,\infty)\subset\mathbb{R}$ $(X,\mathcal{A},\mu)$ is a measure space. The class of all measurable functions for which $\|f\|^p$ is integrable
$L^p$	Measure Theory	Same as $\mathcal{L}^p$
$T_p(A)$ Common: $T_p(\mathbb{R}^n)$	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 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 germs

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

Jump up ↑ John M Lee - Introduction to smooth manifolds - Second edition

[1] Jump up ↑ John M Lee - Introduction to smooth manifolds - Second edition

[1]

@@ Line 58: / Line 58: @@
 The unit interval will be assumed when missing
 |-
-| <math>\mathcal{D}_a(\mathbb{R}^n)</math>
+| <math>D_a(A)</math><br/>Common: <math>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}})
+| Denotes [[Set of all derivations at a point]] - Not to be confused with [[Set of all derivations of a germ]] which is denoted {{M|\mathcal{D}_p(A)}}<br/>
+'''Note:''' This is my/Alec's notation for it, as the author<ref>John M Lee - Introduction to smooth manifolds - Second edition</ref> uses {{M|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]]
+|-
+| <math>\mathcal{D}_a(A)</math><br/>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 {{M|T_p(A)}}
 |-
 | <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
 |-
@@ Line 84: / Line 92: @@
 | Same as <math>\mathcal{L}^p</math>
 |-
-| <math>T_p(\mathbb{R}^n)</math>
+| <math>T_p(A)</math><br/>Common:<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)}}
+Sometimes denoted {{M|\mathbb{R}^n_a}} - '''Note:''' sometimes can mean [[Set of all derivations at a point]] which is denoted {{M|D_a(\mathbb{R}^n)}} and not to be confused with <math>\mathcal{D}_a(\mathbb{R}^n)</math> which denotes [[Set of all derivations of germs]]
 |}

Difference between revisions of "Index of notation"

Revision as of 02:55, 5 April 2015

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