Index of spaces
Using the index
People might use [ilmath]i[/ilmath] or [ilmath]j[/ilmath] or even [ilmath]k[/ilmath] for indicies, as such "numbers" are indexed as "num" (notice the lowercase) so a space like [ilmath]C^k[/ilmath] is under C_num.
We do subscripts first, so [ilmath]A_i^2[/ilmath] would be under A _num ^num:2
When breaking up a term into its index key, spaces delimit the blocks, for example [ilmath]L_1^2[/ilmath] 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 [ilmath]C_{1,2} [/ilmath] would become C _num:1+num:2 and the +s are ordered lexicographically.
If there are multiple variable numbers (for example the [ilmath]i[/ilmath] and [ilmath]j[/ilmath] in [ilmath]B_i^j[/ilmath]) we use num for each of them. Even if they're the same (eg both [ilmath]i[/ilmath]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 [ilmath]L(X,Y)[/ilmath] (where [ilmath]X[/ilmath] and [ilmath]Y[/ilmath] are objects (vector spaces, or Banach spaces... ) we use the key obj for these. So [ilmath]L(X,Y)[/ilmath] becomes L ( obj obj )
Ordering
 First come actual numbers.
 Next come num terms.
 Then come infty (which denotes [ilmath]\infty[/ilmath]
 Then comes objects
 Then come letters (upper case  shown as nonitalic uppercase in the index)
 Then come letters (lowercase  shown as capital italics in the index)
 Then come special lowercase letters (shown as capital italics again in the index, with a ! prefixing the name.
 Then come brackets ( first, then [ then {
 Then comes subscript, then comes superscript.
For example [ilmath]C_0[/ilmath] comes before [ilmath]C_i[/ilmath] comes before [ilmath]C_\infty[/ilmath] comes before [ilmath]C_\text{text} [/ilmath].
The space [ilmath]\ell_2[/ilmath] is !L _num:2, and [ilmath]l_2[/ilmath] is L _num:2 which comes before [ilmath]\ell_2[/ilmath]
Index
Space or name  Index  Type  Argument types  Context  Meaning 

[ilmath]C_k\text{ on }U[/ilmath]  C _num ON obj  Class  [ilmath]U[/ilmath]  open set of [ilmath]\mathbb{R}^n[/ilmath] 

(SEE Classes of continuously differentiable functions)  a function is [ilmath]C_k[/ilmath] on [ilmath]U[/ilmath] if [ilmath]U\subset\mathbb{R}^n[/ilmath] is open and the partial derivatives of [ilmath]f:U\rightarrow\mathbb{R}^m[/ilmath] of all orders (up to and including [ilmath]k[/ilmath]) are continuous on [ilmath]U[/ilmath] 
[ilmath]C_k(U)[/ilmath]  C _num ( obj )  Class  [ilmath]U[/ilmath]  open set of [ilmath]\mathbb{R}^n[/ilmath] 

(SEE Classes of continuously differentiable functions)  denotes a set, given [ilmath]U\subseteq\mathbb{R}^n[/ilmath] (that's open) [ilmath]f\in C_k(U)[/ilmath] if [ilmath]f:U\rightarrow\mathbb{R} [/ilmath] has continuous partial derivatives of all orders up to and including [ilmath]k[/ilmath] on [ilmath]U[/ilmath] 
[ilmath]L(X,Y)[/ilmath]  L ( obj obj )  Normed vector space  [ilmath]X[/ilmath], [ilmath]Y[/ilmath]  normed vector spaces 

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 [ilmath]\mathcal{L}(V,W)[/ilmath] denotes all linear maps between [ilmath]L[/ilmath] and [ilmath]W[/ilmath]  this needs investigation 
[ilmath]l_2[/ilmath]  L _num:2  inner product space 

Space of squaresummable sequences 