Difference between revisions of "Index of spaces"
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People might use {{M|i}} or {{M|j}} or even {{M|k}} for indicies, as such "numbers" are indexed as "num" (notice the lower-case) so a space like {{M|C^k}} is under {{C|C_num}}. | People might use {{M|i}} or {{M|j}} or even {{M|k}} for indicies, as such "numbers" are indexed as "num" (notice the lower-case) so a space like {{M|C^k}} is under {{C|C_num}}. | ||
− | We do subscripts first, so {{M|A_i^2}} would be under {{ | + | We do subscripts first, so {{M|A_i^2}} would be under {{C|A _num ^num:2}} |
+ | When breaking up a term into its index key, spaces delimit the blocks, for example {{M|L_1^2}} becomes {{C|L _num:1 ^num:2}} (the subscript comes first, we sort by subscript, then by superscript) | ||
+ | |||
+ | {{C|+}} is used to extend the index keys, for example {{M|C_{1,2} }} would become {{C|C _num:1+num:2}} and the {{C|+}}s are ordered lexicographically. | ||
+ | |||
+ | If there are multiple variable numbers (for example the {{M|i}} and {{M|j}} in {{M|B_i^j}}) we use {{C|num}} for each of them. Even if they're the same (eg both {{M|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. | ||
+ | |||
+ | {{C|*}} denotes objects, so for example say in {{M|L(X,Y)}} (where {{M|X}} and {{M|Y}} are objects (vector spaces, or Banach spaces... ) we use the key {{C|obj}} for these. So {{M|L(X,Y)}} becomes {{C|L ( obj obj )}} | ||
===Ordering=== | ===Ordering=== | ||
# First come actual numbers. | # First come actual numbers. | ||
# Next come {{C|num}} terms. | # Next come {{C|num}} terms. | ||
# Then come {{C|infty}} (which denotes {{M|\infty}} | # Then come {{C|infty}} (which denotes {{M|\infty}} | ||
− | # Then come letters (upper case) | + | # 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 {{C|!}} prefixing the name. | ||
# Then come brackets {{C|(}} first, then {{C|[}} then {{C|{}} | # Then come brackets {{C|(}} first, then {{C|[}} then {{C|{}} | ||
+ | # Then comes subscript, then comes superscript. | ||
− | For example {{M|C_0}} comes before {{M|C_i}} comes before {{M|C_\infty}} comes before {{M|C_\text{text} }} | + | For example {{M|C_0}} comes before {{M|C_i}} comes before {{M|C_\infty}} comes before {{M|C_\text{text} }}. |
+ | |||
+ | The space {{M|\ell_2}} is {{C|!''L'' _num:2}}, and {{M|l_2}} is {{C|''L'' _num:2}} which comes before {{M|\ell_2}} | ||
==Index== | ==Index== | ||
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! Space or name | ! Space or name | ||
! Index | ! Index | ||
+ | ! Type | ||
+ | ! Argument types | ||
! Context | ! Context | ||
! Meaning | ! Meaning | ||
|- | |- | ||
| {{M|C_k\text{ on }U}} | | {{M|C_k\text{ on }U}} | ||
− | ! | + | ! {{nowrap|C _num ON obj}} |
+ | | Class | ||
+ | | {{M|U}} - open set of {{M|\mathbb{R}^n}} | ||
| | | | ||
* ''(Everywhere)'' | * ''(Everywhere)'' | ||
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|- | |- | ||
| {{M|C_k(U)}} | | {{M|C_k(U)}} | ||
− | ! | + | ! {{nowrap|C _num ( obj )}} |
+ | | Class | ||
+ | | {{M|U}} - open set of {{M|\mathbb{R}^n}} | ||
| | | | ||
* ''(Everywhere)'' | * ''(Everywhere)'' | ||
| '''(SEE ''[[Classes of continuously differentiable functions]]'')''' - denotes a set, given {{M|U\subseteq\mathbb{R}^n}} (that's open) {{M|f\in C_k(U)}} if {{M|f:U\rightarrow\mathbb{R} }} has continuous partial derivatives of all orders up to and including {{M|k}} on {{M|U}} | | '''(SEE ''[[Classes of continuously differentiable functions]]'')''' - denotes a set, given {{M|U\subseteq\mathbb{R}^n}} (that's open) {{M|f\in C_k(U)}} if {{M|f:U\rightarrow\mathbb{R} }} has continuous partial derivatives of all orders up to and including {{M|k}} on {{M|U}} | ||
+ | |- | ||
+ | | {{M|L(X,Y)}} | ||
+ | ! {{nowrap|L ( obj obj )}} | ||
+ | | Normed vector space | ||
+ | | {{M|X}}, {{M|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 {{M|\mathcal{L}(V,W)}} denotes all linear maps between {{M|L}} and {{M|W}} - this needs investigation}} | ||
|- | |- | ||
| {{M|l_2}} | | {{M|l_2}} | ||
− | ! | + | ! ''L'' _num:2 |
+ | | inner product space | ||
+ | | | ||
| | | | ||
* Functional Analysis | * Functional Analysis |
Latest revision as of 21:06, 29 February 2016
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 |