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Difference between pages "Talk:Unbounded operator" and "User:Boris Tsirelson"

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(I didn't say anything of the kind)
 
m (typo)
 
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"The simplest example of an unbounded operator is the differentiation operator $ \dfrac{\mathrm{d}}{\mathrm{d}{t}} $, defined on the set $ {C^{1}}([a,b]) $ of all continuously differentiable functions into the space $ C([a,b]) $ of all continuous functions on $ a \leq t \leq b $, because the operator $ \dfrac{\mathrm{d}}{\mathrm{d}{t}} $ takes the bounded set $ \{ t \mapsto \sin(n t) \}_{n \in \mathbb{N}} $ to the unbounded set $ \{ t \mapsto n \cos(n t) \}_{n \in \mathbb{N}} $" — really? The set $ \{ t \mapsto \sin(n t) \}_{n \in \mathbb{N}} $ is unbounded in the space $ {C^{1}}([a,b]) $; and the operator is bounded from the space $ {C^{1}}([a,b]) $ to the space $ C([a,b]) $. True, the article mentions " the set $ {C^{1}}([a,b]) $", not  "the space $ {C^{1}}([a,b]) $"; but it is not said that this set is treated as a subset of the space $ C([a,b]) $. The text may be misleading. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 06:52, 1 March 2017 (CET)
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My [http://www.tau.ac.il/~tsirel/ homepage] at School of Mathematics, Tel Aviv University.
  
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:Hi Boris. Yes, I agree that the original text is misleading. I can change ‘set’ to ‘space’ so that $ ': {C^{1}}([a,b]) \to C([a,b]) $ is an unbounded linear operator from one normed vector space to another. I must admit that I’m not entirely happy with the definition of ‘unbounded operator’ being offered here. Standard practice, I believe, is to call any densely defined linear operator $ S $ from one normed vector space $ X $ to another $ Y $ an ‘unbounded linear operator’ even if $ S $ has a bounded extension $ T: X \to Y $. [[User:Leonard Huang|Leonard Huang]] ([[User talk:Leonard Huang|talk]]) 07:27, 3 March 2017 (CET)
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My [http://en.wikipedia.org/wiki/User:Tsirel user page] on Wikipedia.
  
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::Really? I did not know. But wait, the whole $X$ is also a dense linear subset of $X$. Do you mean that (usual, defined everywhere) bounded operators are a special case of unbounded operators? [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 09:15, 3 March 2017 (CET)
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My research interests are in [[:Category:Probability theory and stochastic processes|probability]].
  
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:::Hi Boris. Yes, that is (unfortunately) so — a bounded globally defined linear operator between normed vector spaces (Banach spaces or Hilbert spaces mostly) is, by definition, an unbounded linear operator. The [https://ncatlab.org/nlab/show/unbounded+operator nLab article on unbounded operators] calls this an example of a red herring. [[User:Leonard Huang|Leonard Huang]] ([[User talk:Leonard Huang|talk]]) 11:37, 3 March 2017 (CET)
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* ''Boris has [https://en-exact-sciences.tau.ac.il/math/Tsirelsonborisen passed away] on January 21, 2020. May he rest in peace.''
  
−
::::I see. Well, it does not upset me. I've read with pleasure [https://ncatlab.org/nlab/show/red+herring+principle#examples red herring], and I can add more examples: a finitely additive measure is generally not a measure, but every measure is a finitely additive measure; etc. Let us change the article accordingly. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 12:02, 3 March 2017 (CET)
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==My contributions==
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::::And, by the way, [https://en.wikipedia.org/wiki/Unbounded_operator Wikipedia says]: "The term "unbounded operator" can be misleading, since
 
−
::::*"unbounded" should sometimes be understood as "not necessarily bounded";
 
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::::*"operator" should be understood as "linear operator" (as in the case of "bounded operator");
 
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::::*the domain of the operator is a linear subspace, not necessarily the whole space;
 
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::::*this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense;
 
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::::*in the special case of a bounded operator, still, the domain is usually assumed to be the whole space."
 
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::::[[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 18:39, 4 March 2017 (CET)
 
  
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:::::''Please'' let's not base anything on what Wikipedia says.  Seriously.  If there's a point at issue here, let's get someone who knows the area well to advise.  [[User:Richard Pinch|Richard Pinch]] ([[User talk:Richard Pinch|talk]]) 08:41, 5 March 2017 (CET)
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[[Measurable space]] (50,000+ views);
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[[Standard Borel space]] (12,000+ views);
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[[Analytic Borel space]] (5,000+ views);
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[[Universally measurable]] (5,000+ views);
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[[Measure space]] (20,000+ views);
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[[Standard probability space]] (6,000+ views);
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[[Measure algebra (measure theory)]] (7,000+ views).
  
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::::::That is, you feel unsure about "the standard practice", do you? [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 12:30, 5 March 2017 (CET)
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===Wanted===
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:::::::I don't know what you mean by that, but I do not think it is related to what I said. Please explain. [[User:Richard Pinch|Richard Pinch]] ([[User talk:Richard Pinch|talk]]) 21:54, 6 March 2017 (CET)
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::::::::Sorry for being unclear. Yes, I mean your phrase "Standard practice, I believe, is to call any densely defined linear operator $ S $ from one normed vector space $ X $ to another $ Y $ an ‘unbounded linear operator’ even if $ S $ has a bounded extension $ T: X \to Y $" above. As for me, your phrase and the quote from wikipedia conform. However, you seem to object to the latter. I wonder, why. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 21:16, 7 March 2017 (CET)
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[[Random matrix]]
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:::::::::I didn't say anything of the kind – [[User:Leonard Huang|Leonard Huang]] did, though – so your phrase "your phrase" remains unclear. I object to Wikipedia in principle: it is not a reliable source. [[User:Richard Pinch|Richard Pinch]] ([[User talk:Richard Pinch|talk]]) 23:15, 7 March 2017 (CET)
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===Know-how===
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Symbol choice: [[Talk:Variance]], [[Talk:Almost-periodic analytic function]], [[Talk:Borel-Cantelli lemma]].
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<nowiki>#REDIRECT[[]]</nowiki>
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<nowiki>{{Anchor|countably separated}}</nowiki>
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<nowiki>$\int f(x)\rd x$</nowiki> &nbsp; $\int f(x)\rd x$
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<nowiki>$\int f(x)\,\mu(\!\rd x)$</nowiki> &nbsp; $\int f(x)\,\mu(\!\rd x)$
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<nowiki>See {{Cite|K|Sect. 12.B}}.</nowiki>
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<nowiki>{|</nowiki><br />
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<nowiki>|valign="top"|{{Ref|K}}|| Alexander S.  Kechris, "Classical  descriptive set theory", Springer-Verlag (1995). &amp;nbsp;  {{MR|1321597}} &amp;nbsp; {{ZBL|0819.04002}</nowiki><br />
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<nowiki>|-</nowiki><br />
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<nowiki>|valign="top"|{{Ref|M}}||  George W. Mackey, "Borel structure in groups and their duals",  ''Trans.  Amer.  Math. Soc.''  '''85''' (1957), 134–165. &amp;nbsp; {{MR|0089999}} &amp;nbsp; {{ZBL|0082.11201}}</nowiki><br />
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<nowiki>|}</nowiki>
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====Article====
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''Also: standard measurable space''
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<nowiki>{{MSC|03E15|28A05,54H05}}</nowiki>
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<nowiki>[[Category:Descriptive set theory]]</nowiki><br>
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<nowiki>[[Category:Classical measure theory]]</nowiki>
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<nowiki>$ \newcommand...$</nowiki><br>
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A Borel space is called '''standard''' if...
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-----------------------------------------------------
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<nowiki>{{MSC|60-01|60F15,60F20}}</nowiki>
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-----------------------------------------------------
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<nowiki>{{DEF}}</nowiki><br>
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This is a "wanted page". It was proposed that an entry with this title should be included.<br>
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<nowiki>{{TEX|want}}</nowiki><br>
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''subtitle''<br>
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<nowiki>{{MSC|...}}</nowiki><br>
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<nowiki>[[Category:...]]</nowiki><br>
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<nowiki>{{TEX|done}}</nowiki><br>
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====Category====
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<nowiki>* {{MSC|60J10}}</nowiki>
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<nowiki>[[Category:Markov processes]]</nowiki>
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--------------------------------------
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<nowiki>* {{MSC|60Jxx}}</nowiki>
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<nowiki>[[Category:Probability theory and stochastic processes]]</nowiki>
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--------------------------------------
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<nowiki>* {{MSC|60}}</nowiki>
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<nowiki>[[Category:Probability and statistics]]</nowiki>
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====Talk====
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: And please sign your messages (on talk pages) with four tildas: <nowiki>~~~~</nowiki>.

Latest revision as of 05:59, 19 January 2022

My homepage at School of Mathematics, Tel Aviv University.

My user page on Wikipedia.

My research interests are in probability.

  • Boris has passed away on January 21, 2020. May he rest in peace.

My contributions

Measurable space (50,000+ views); Standard Borel space (12,000+ views); Analytic Borel space (5,000+ views); Universally measurable (5,000+ views); Measure space (20,000+ views); Standard probability space (6,000+ views); Measure algebra (measure theory) (7,000+ views).

Wanted

Random matrix + SLE + Gaussian measure

And maybe: Borel equivalence relation (see WP)

Require revision

Bell inequalities + Probability of large deviations

Some statistics

We have 2590+ articles viewed 2590+ times each. [1]

History: Dec 2011: 40×40; Jan 2012: 60×60; Feb 2012: 90×90; Mar 2012: 140×140; Apr 2012: 200×200; May 2012: 250×250; Jun 2012: 320×320; Jul 2012: 370×370; Aug 2012: 410×410; Sep 2012: 460×460; Oct 2012: 520×520; Nov 2012: 590×590; Dec 2012: 650×650; Jan 2013: 710×710; Feb 2013: 760×760; Mar 2013: 820×820; Apr 2013: 890×890; May 2013: 940×940; Jun 2013: 990×990; Jul 2013: 1040×1040; Aug 2013: 1080×1080; Sep 2013: 1130×1130; Oct 2013: 1170×1170; Nov 2013: 1220×1220; Dec 2013: 1260×1260; Jan 2014: 1300×1300; Feb 2014: 1330×1330; Mar 2014: 1380×1380; Apr 2014: 1420×1420; May 2014: 1480×1480; Jun 2014: 1520×1520; Jul 2014: 1600×1600; Aug 2014: 1640×1640; Sep 2014: 1700×1700; Oct 2014: 1760×1760; Nov 2014: 1800×1800; Dec 2014: 1860×1860; Jan 2015: 1920×1920; Feb 2015: 1960×1960; Mar 2015: 2020×2020; Apr 2015: 2080×2080; May 2015: 2140×2140; June 2015: 2180×2180; Aug 2015: 2260×2260; Sep 2015: 2280×2280; Oct 2015: 2290×2290; Nov 2015: 2300×2300; Dec 2015: 2310×2310; Jan 2016: 2320×2320; Feb 2016: 2340×2340; Jun 2016: 2400×2400; Oct 2016: 2480×2480; Feb 2017: 2590x2590.

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Notes to myself

Measurable space: 48149 views at 14 November 2015; 49,116 views at 1 July 2016; 52,310 views at 7 March 2017. Rigid analytic space: 2298 views at 14 November 2015; 2,454 views at 1 July 2016; 2,749 views at 7 March 2017.

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Know-how

Symbol choice: Talk:Variance, Talk:Almost-periodic analytic function, Talk:Borel-Cantelli lemma.

#REDIRECT[[]]

{{Anchor|countably separated}}

$\int f(x)\rd x$   $\int f(x)\rd x$

$\int f(x)\,\mu(\!\rd x)$   $\int f(x)\,\mu(\!\rd x)$

See {{Cite|K|Sect. 12.B}}.

{|
|valign="top"|{{Ref|K}}|| Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995). &nbsp; {{MR|1321597}} &nbsp; {{ZBL|0819.04002}
|-
|valign="top"|{{Ref|M}}|| George W. Mackey, "Borel structure in groups and their duals", ''Trans. Amer. Math. Soc.'' '''85''' (1957), 134–165. &nbsp; {{MR|0089999}} &nbsp; {{ZBL|0082.11201}}
|}


Article

Also: standard measurable space

{{MSC|03E15|28A05,54H05}}

[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]

$ \newcommand...$
A Borel space is called standard if...

{{MSC|60-01|60F15,60F20}}

{{DEF}}
This is a "wanted page". It was proposed that an entry with this title should be included.
{{TEX|want}}
subtitle
{{MSC|...}}
[[Category:...]]
{{TEX|done}}

Category

* {{MSC|60J10}}

[[Category:Markov processes]]

* {{MSC|60Jxx}}

[[Category:Probability theory and stochastic processes]]

* {{MSC|60}}

[[Category:Probability and statistics]]

Talk

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How to Cite This Entry:
Unbounded operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unbounded_operator&oldid=40229
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