# 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 lower-case) 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

1. First come actual numbers.
2. Next come num terms.
3. Then come infty (which denotes [ilmath]\infty[/ilmath]
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 [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]
• (Everywhere)
(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]
• (Everywhere)
(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
• 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 [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
• Functional Analysis
Space of square-summable sequences