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I agree, really. The edit from last June which introduced the definition by symbols is not really in the right place. It should go after the more verbose definition. Also the first paragraph should be expanded with some gentle introduction to the general idea: the norm of an operator gives us a
175:
What could maybe make this article better is to split it into two articles. One main one about linear bounded operators where one focuses on linear operators and not as much on fine details about their norms, and a shorter article on operator norms, which could be more techical.
187:
specific way to talk about its 'size', operator norms apply to matrices but also are particularly useful in picturing the case of spaces of infinite dimension, if the operator norm fails to be defined because the sup is unbounded that indicates the operator is not continuous.
229:
Just a minor thing: the second section says that $ \{c : \|Av\| \leq c\|v\| \forall v \in V\}$ may have no minimum, but I dispute that. It's the intersection over all nonzero $ v\in V$ of the closed set $ $ , and an abritrary intersection of closed sets is closed.
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Except for the NP-hard norms, all these norms can be calculated in N^2 operations (for a NxN matrix), with the exception of the l2-l2 norm (which requires N^3 operations for the exact answer, or less if you approximate it with the power method or
Lanczos iterations).
268:. This is a bit confusing. Is it possible to coordinate this information so both of the articles give the same information? Also, the information given in the latter article, does not seem to correspond to the information given in the WolframMathWorld articles about
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Looking down below, one uses the words "bounded linear transformation" to introduce the notion of norm. Some care is needed here, as a bounded linear operator is not implying it to be a bounded linear function.
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298:. I think this might make a nice addition to the main article. (This could be made more complete by discussing which of the NP-hard norms allow quick approximations).
216:, as these go hand in hand. However, the way things are now is not so good, so two articles, even with a small amount of repetition among them, should be better.
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it says that the induced norm is the same as the operator norm. Then one definition is given, which is not the same definition given in this article about
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Some common operator norms are easy to calculate, and others are NP-hard. There's a very nice table in section 4.3.1 of Joel Tropp's PhD thesis from 2004
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redirects here. Well, that is a self-link that needs to be fixed. So, Oleg's idea maybe to split that into a separate article seems quite promising.
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First, the article starts with a very technical "all operator norm definitions money can buy", which is certainly not encouraging.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Also, the first section does not tie in well with the remainder of the article in general.
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with standard inner product, then we're forced to use the 2 norm, not the 1 norm.
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I think the resolution is that your example isn't in
Hilbert space! With
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I will get to it by this weeked. It would be not easy to separate
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Inconsistency between operator norm and induced norm article
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Maybe I'm missing something, but this passage seems wrong:
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Sorry for any mistakes and thanks for any clarifications.
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115:Knowledge:WikiProject Mathematics
118:Template:WikiProject Mathematics
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903:{\displaystyle \|A^{*}\|_{1}=7}
590:{\displaystyle \ell _{\infty }}
398:{\displaystyle \ell _{\infty }}
306:Computability of Operator Norm
135:This article has been rated as
291:Table of common operator norms
286:15:07, 12 September 2010 (UTC)
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109:and see a list of open tasks.
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857:{\displaystyle \|A\|_{1}=6}
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625:{\displaystyle \ell _{1}}
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139:on the
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