Index of notation

[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]	Functional Analysis Real Analysis	Denotes the Norm of a vector
[math]\|f\|_{C^k}[/math]	Functional Analysis	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]	Functional Analysis	[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]	Functional Analysis 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 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]	Differential Geometry Manifolds	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]	Differential Geometry Manifolds	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]]	Differential Geometry Manifolds	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]	Differential Geometry Manifolds	[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] The set of all germs of smooth functions at a point
[math]C^k([a,b],\mathbb{R})[/math]	Functional Analysis Real Analysis	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]	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 [ilmath]\mathcal{D}_p(A)[/ilmath] Note: This is my/Alec's notation for it, as the author 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]	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 [ilmath]T_p(A)[/ilmath]	TANGENT
[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
[math]G_p(\mathbb{R}^n)[/math]	Differential Geometry Manifolds	The geometric tangent space - see Geometric Tangent Space	TANGENT_NEW
[math]\ell^p(\mathbb{F})[/math]	Functional Analysis	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]	Measure Theory	[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]L^p[/math]	Measure Theory	Same as [math]\mathcal{L}^p[/math]
[math]T_p(A)[/math] Common:[math]T_p(\mathbb{R}^n)[/math]	Differential Geometry Manifolds	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	Measure Theory	There exists a Measurable map between the [ilmath]\sigma[/ilmath]-algebras
[ilmath]a\cdot b[/ilmath]	Anything with vectors	Vector dot product