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basis without essentially using AC for some cardinal. However, there is no reason why it should say Zorn's lemma instead of the axiom of choice, and the point is largely irrelevant for functional analysis, as there are very few cases where one actually needs a Hamel basis for a particular infinite-dimensional Banach space.
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3. "For any real number p ≥ 1, an example of a Banach space is given by "all
Lebesgue-measurable functions whose absolute value's p-th power has finite integral" (see Lp spaces)" ("Banch spaces" paragraph) Since this sentence is supposed to present an example of Banch space for a general audience, I
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5."geometry of Banach spaces, a combinatorial approach as in the work of Jean
Bourgain" (in "Points of view"). The study of geometry of Banach spaces requires much more than combinatorics. Also, the study was initiated by Fritz-Johnes and Dvoretzky work and then developed into "theory" by works of
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I fixed the remark saying that Hahn-Banach requires AC. Hahn-Banach can only be side-stepped in separable spaces if one allows DC. The comment about infinite-dimensional spaces is correct, I would think, because almost all spaces actually considered in functional analysis can not be shown to have a
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There are two versions, with ISBNs 978-2100493364 and 978-2100043149. Unfortunately, none of the book finder links in wikipedia's book finder seem to work, and it's not on amazon.com, but both versions may be found at amazon.ca, and the former may be found at amazon.fr. Go figure. However, I still
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2. "There is no clear definition of what would constitute a base, for example" (very next sentence). This is definetely not true. The notion of a base in Banach spaces is well defined and understood, see any text book on
Functional Analysis. Moreover, this sentence seems to relate somehow to its
118:. I'm sure much of this is wrong, and much of the rest is POV. Some infinite-dimensional spaces can be shown to have a basis without AC, surely. Hahn-Banach can be side-stepped in separable spaces. And the advice is not necessarily good. Is any of this really important to keep?
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Functional
Analysis is also a well known term in Applied Behavior Analysis (a branch of psychology) and concerns the analysis of the function of a particular behavior, especially problem or maladaptive behaviors. I would suggest a cross reference to Applied Behavior Analysis.
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I am unaware of a singel case of a Banch space which requires use of Zorn's lemma to show that it has a basis.Usually, a basic sequence is "more or less a natural choice" and the prove is focused on showing that, indeed, the sequence forms a basis.
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In the article it was said that "every operator on a
Hilbert space has a proper subspace which is invariant". Now is invariant some functional theoretic term or is that invariant under some transformation? I'm not an expert on FA.
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1."General Banach spaces are more complicated." (in "Banach spaces" paragraph) I assume that this relates Hilber and Banach spaces. This is a type of "apples or bananas" issue and there is no need to try to resolve it here.
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V.Milman (and others). I am not trying to take anything from outstanding work of J. Bourgain, I am just saying that this part of the article contains two misleading references.
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208:. By the way, you should not use the minor edit checkmark for edits which are not minor, like for example starting a new topic on talk pages. Also, it is good to use
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would suggest that l^p (or C) be used instead of L^p. I mean, one would not have to invoke concept of
Lebesgue mesaurable functions.
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predicessor and suggests that Banch spaces are so difficult to understand that people do not even know how to define a base.
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2."The Volume of Convex Bodies and Banach Space
Geometry " by by Gilles Pisier,ISBN: 0521364655 ,Cambridge University Press
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6. References section. It contains references to books at least 50 years old. I suggest that the following be added:
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1. "Classical Banach spaces" by Joram
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4."To show the existence of a vector space basis for such spaces may require Zorn's lemma"
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I have a few comments about wording and substance of a few claims raised in this article:
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Is it really kind to give the three volumes of
Dunford and Schwartz as basic reference?
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That's not completely true. In fact, very much is unknown even about the geometry of
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If you wish to start a new discussion or revive an old one, please do so on the
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Analysis is also an important part of Applied Behavior Analysis
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The following discussion relates to a now abandoned proposal to
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250:(Second sentence in "Foundation of mathematics" paragraph)
99:, my man, these subjects diverged nearly a century ago.
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212:, so that others can tell what you changed. Thanks.
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