Index of spaces

Using the index

People might use $i$ or $j$ or even $k$ for indicies, as such "numbers" are indexed as "num" (notice the lower-case) so a space like $C^k$ is under C_num.

We do subscripts first, so $A_i^2$ would be under A _num ^num:2

When breaking up a term into its index key, spaces delimit the blocks, for example $L_1^2$ becomes L _num:1 ^num:2 (the subscript comes first, we sort by subscript, then by superscript)

+ is used to extend the index keys, for example $C_{1,2}$ would become C _num:1+num:2 and the +s are ordered lexicographically.

If there are multiple variable numbers (for example the $i$ and $j$ in $B_i^j$) we use num for each of them. Even if they're the same (eg both $i$s or something) - while not ideal the index should be small enough (when you've got a leading letter) that you do not need any further granularity.

* denotes objects, so for example say in $L(X,Y)$ (where $X$ and $Y$ are objects (vector spaces, or Banach spaces... ) we use the key obj for these. So $L(X,Y)$ becomes L ( obj obj )

Ordering

1. First come actual numbers.
2. Next come num terms.
3. Then come infty (which denotes $\infty$
4. Then comes objects
5. Then come letters (upper case - shown as non-italic uppercase in the index)
6. Then come letters (lower-case - shown as capital italics in the index)
7. Then come special lowercase letters (shown as capital italics again in the index, with a ! prefixing the name.
8. Then come brackets ( first, then [ then {
9. Then comes subscript, then comes superscript.

For example $C_0$ comes before $C_i$ comes before $C_\infty$ comes before $C_\text{text}$.

The space $\ell_2$ is !L _num:2, and $l_2$ is L _num:2 which comes before $\ell_2$

Index

Space or name	Index	Type	Argument types	Context	Meaning
$C_k\text{ on }U$	C _num ON obj	Class	$U$ - open set of $\mathbb{R}^n$	(Everywhere)	(SEE Classes of continuously differentiable functions) - a function is $C_k$ on $U$ if $U\subset\mathbb{R}^n$ is open and the partial derivatives of $f:U\rightarrow\mathbb{R}^m$ of all orders (up to and including $k$) are continuous on $U$
$C_k(U)$	C _num ( obj )	Class	$U$ - open set of $\mathbb{R}^n$	(Everywhere)	(SEE Classes of continuously differentiable functions) - denotes a set, given $U\subseteq\mathbb{R}^n$ (that's open) $f\in C_k(U)$ if $f:U\rightarrow\mathbb{R}$ has continuous partial derivatives of all orders up to and including $k$ on $U$
$L(X,Y)$	L ( obj obj )	Normed vector space	$X$, $Y$ - normed vector spaces	Analysis Functional analysis Linear algebra	It's the Space of all continuous linear functions between two normed vector spaces and it itself is a normed vector space. Warning:I'm not sure if this differs or is universal, there can be discontinuous linear maps between spaces, however another book tells me $\mathcal{L}(V,W)$ denotes all linear maps between $L$ and $W$ - this needs investigation
$l_2$	L _num:2	inner product space		Functional Analysis	Space of square-summable sequences