Difference between revisions of "Index of spaces"

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(Created page with "{| class="wikitable" border="1" |- ! Index ! Space ! Context ! Meaning |- ! L2 | {{M|l_2}} | * Functional Analysis | Space of square-summable sequences |} Category:Ind...")
 
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==Using the index==
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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}}.
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We do subscripts first, so {{M|A_i^2}} would be under {{C|A _num ^num:2}}
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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)
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{{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.
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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.
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{{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 )}}
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===Ordering===
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# First come actual numbers.
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# Next come {{C|num}} terms.
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# Then come {{C|infty}} (which denotes {{M|\infty}}
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# Then comes objects
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# Then come letters (upper case - shown as non-italic uppercase in the index)
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# Then come letters (lower-case - shown as capital italics in the index)
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# Then come special lowercase letters (shown as capital italics again in the index, with a {{C|!}} prefixing the name.
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# Then come brackets {{C|(}} first, then {{C|[}} then {{C|{}}
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# Then comes subscript, then comes superscript.
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For example {{M|C_0}} comes before {{M|C_i}} comes before {{M|C_\infty}} comes before {{M|C_\text{text} }}.
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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}}
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==Index==
 
{| class="wikitable" border="1"
 
{| class="wikitable" border="1"
 
|-
 
|-
 +
! Space or name
 
! Index
 
! Index
−
! Space
+
! Type
 +
! Argument types
 
! Context
 
! Context
 
! Meaning
 
! Meaning
 
|-
 
|-
−
! L2
+
| {{M|C_k\text{ on }U}}
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! {{nowrap|C _num ON obj}}
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| Class
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| {{M|U}} - open set of {{M|\mathbb{R}^n}}
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|
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* ''(Everywhere)''
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| '''(SEE ''[[Classes of continuously differentiable functions]]'')''' - a function is {{M|C_k}} on {{M|U}} if {{M|U\subset\mathbb{R}^n}} is open and the partial derivatives of {{M|f:U\rightarrow\mathbb{R}^m}} of all orders (up to and including {{M|k}}) are continuous on {{M|U}}
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|-
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| {{M|C_k(U)}}
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! {{nowrap|C _num ( obj )}}
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| Class
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| {{M|U}} - open set of {{M|\mathbb{R}^n}}
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|
 +
* ''(Everywhere)''
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| '''(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}}
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|-
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| {{M|L(X,Y)}}
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! {{nowrap|L ( obj obj )}}
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| Normed vector space
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| {{M|X}}, {{M|Y}} - normed vector spaces
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|
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* Analysis
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* Functional analysis
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* Linear algebra
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| 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}}
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|-
 
| {{M|l_2}}
 
| {{M|l_2}}
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! ''L'' _num:2
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| inner product space
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|
 
|
 
|
 
* 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

  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
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