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
- 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 non-italic uppercase in the index)
- Then come letters (lower-case - 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 square-summable sequences |
