Index of notation

$$\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }$$ $$\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}$$ $$\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }$$ 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 $$C^k$$ [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 $$C^\infty$$ 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 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	$$\mathcal{L}^p(\mu)=\{u:X\rightarrow\mathbb{R}|u\in\mathcal{M},\ \int|u|^pd\mu<\infty\},\ p\in[1,\infty)\subset\mathbb{R}$$ $$(X,\mathcal{A},\mu)$$ is a measure space. The class of all measurable functions for which $$|f|^p$$ is integrable
$$\mathcal{L}(V,W)$$	Linear Algebra	The set of all linear maps from a vector space $V$ (over a field $F$) and another vector space $W$ also over $F$. It is a vector space itself. See The vector space of all maps between vector spaces
$$\mathcal{L}(V)$$	Linear algebra	Short hand for $$\mathcal{L}(V,V)$$ (see above). In addition to being a vector space it is also an Algebra
$$L^p$$	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