Difference between revisions of "Index of notation"

Revision as of 23:53, 16 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.

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

$A$ comes before

$\mathbb{A}$ comes before

$\mathcal{A}$

Expression	Context	Details	Mark
$\\|\cdot\\|$	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 See also Smooth function and the symbols $C^\infty(\mathbb{R}^n)$ and $C^\infty(M)$ where $M$ is a Smooth manifold
$C^\infty(\mathbb{R}^n)$	Differential Geometry Manifolds	The set of all Smooth functions on $\mathbb{R}^n$ - see Smooth function, it means $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is Smooth in the usual sense - all partial derivatives of all orders are continuous.	TANGENT_NEW
$C^\infty(M)$	Differential Geometry Manifolds	The set of all Smooth functions on the Smooth manifold $M$ - see Smooth function, it means $f:M\rightarrow\mathbb{R}$ is smooth in the sense defined on Smooth function	TANGENT_NEW
$C^k$ [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	$C^\infty_p(A)$ 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	TANGENT
$\mathcal{D}_a(A)$ 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 $T_p(A)$	TANGENT
$\bigudot_i A_i$	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
$G_p(\mathbb{R}^n)$	Differential Geometry Manifolds	The geometric tangent space - see Geometric Tangent Space	TANGENT_NEW
$\ell^p(\mathbb{F})$	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 $\mathcal{L}^p$
$\mathbb{S}^n$	Real Analysis Differential Geometry Manifolds Topology	$\mathbb{S}^n\subset\mathbb{R}^{n+1}$ and is the $n$ -sphere, examples: $\mathbb{S}^1$ is a circle, $\mathbb{S}^2$ is a sphere, $\mathbb{S}^0$ is simply two points.
$T_p(A)$ 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 a germ	TANGENT

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
$p_0\simeq p_1\text{ rel}\{0,1\}$	Topology	See Homotopic paths

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 177: / Line 177: @@
 * Anything with vectors
 | [[Vector dot product]]
+|-
+| <math>p_0\simeq p_1\text{ rel}\{0,1\}</math>
+|
+* Topology
+| See [[Homotopic paths]]
 |}

Difference between revisions of "Index of notation"

Revision as of 23:53, 16 April 2015

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