View source for Talk:Unbounded operator

"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)

: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)

::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)

:::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)

::::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)
::::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";
::::*"operator" should be understood as "linear operator" (as in the case of "bounded operator");
::::*the domain of the operator is a linear subspace, not necessarily the whole space;
::::*this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense;
::::*in the special case of a bounded operator, still, the domain is usually assumed to be the whole space."
::::[[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 18:39, 4 March 2017 (CET)

:::::''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)

::::::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)
:::::::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)
::::::::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)
:::::::::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)
::::::::::Oops! I am sorry for my inadvertence. Now I understand. Yes, you are right, in general. But in particular (revealing the secret) those Wikipedia phrases are written (mostly) by myself. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 06:21, 8 March 2017 (CET)