Wikipedia:WikiProject Mathematics/PlanetMath Exchange/46-XX Functional analysis: Difference between revisions

Revision as of 21:44, 4 March 2006

Revision as of 01:43, 5 March 2006

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==46-00 General reference works (handbooks, dictionaries, bibliographies, etc.)==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4434 T_f is a distribution of zeroth order], id=4434 -- WP: [[Distribution]] -- Status:

::PM article is a proof which is not included in WP; WP doesn't even define the ''order'' of a distribution. [[User:Jitse Niesen|Jitse Niesen]] 13:32, 31 Jan 2005 (UTC)

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:::I recently split the page from [[bounded set]] and added the theorem from PM. [[User:MathMartin|MathMartin]] 11:49, 4 May 2005 (UTC)

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4472 Cauchy principal part integral], id=4472 -- WP guess: [[Cauchy principal part integral]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6195 Cauchy sequence], id=6195 -- WP: [[Cauchy sequence]] -- Status: '''A'''

::Our artilcle is more complete. [[User:Paul August|Paul August ]] [[User_talk:Paul August|☎]] 15:03, Jan 31, 2005 (UTC)

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4570 cone], id=4570 -- WP guess: [[cone]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4468 delta distribution], id=4468 -- WP guess: [[delta distribution]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4427 distribution], id=4427 -- WP guess: [[distribution]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4433 every locally integrable function is a distribution], id=4433 -- WP guess: [[every locally integrable function is a distribution]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=5556 function spaces], id=5556 -- WP guess: [[function spaces]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4477 localization for distributions], id=4477 -- WP guess: [[localization for distributions]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4424 locally convex topological vector space], id=4424 -- WP guess: [[locally convex topological vector space]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4440 operations on distributions], id=4440 -- WP guess: [[operations on distributions]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4437 proof of convergence theorem], id=4437 -- WP guess: [[proof of convergence theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4525 sequential characterization of boundedness], id=4525 -- WP guess: [[sequential characterization of boundedness]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4568 smooth distribution], id=4568 -- WP guess: [[smooth distribution]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4484 support of distribution], id=4484 -- WP guess: [[support of distribution]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4528 symmetric set], id=4528 -- WP guess: [[symmetric set]] -- Status:

==46A03 General theory of locally convex spaces==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=5921 Krein-Milman theorem], id=5921 -- WP: [[Krein-Milman theorem]] -- Status: '''C'''

:: [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 19:16, 20 November 2005 (UTC)

==46A30 Open mapping and closed graph theorems; completeness (including $B$-, $B r$-completeness)==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=3702 closed graph theorem], id=3702 -- WP: [[closed graph theorem]] -- Status: '''A'''

:: [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 19:01, 20 November 2005 (UTC)

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==46A55 Convex sets in topological linear spaces; Choquet theory==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4443 convex hull of S is open if S is open], id=4443 -- WP guess: [[convex hull of S is open if S is open]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=5587 proof that the4443 convex hull of S is open if S is open], id=55874443 -- WP guess: [[proof that the convex hull of S is open if S is open]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=44435587 proof that the convex hull of S is open if S is open], id=44435587 -- WP guess: [[proof that the convex hull of S is open if S is open]] -- Status:

==46A99 Miscellaneous==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=3066 Heine-Cantor theorem], id=3066 -- WP: [[Heine-Cantor theorem]] -- Status: '''M'''

::[[User:AdamSmithee|AdamSmithee]] 15:29, 3 February 2006 (UTC)

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==46B03 Isomorphic theory (including renorming) of Banach spaces==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4315 hyperbolic isomorphism], id=4315 -- WP guess: [[hyperbolic isomorphism]] -- Status:

==46B07 Local theory of Banach spaces==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=5927 Lipschitz inverse mapping theorem], id=5927 -- WP guess: [[Lipschitz inverse mapping theorem]] -- Status:

==46B10 Duality and reflexivity==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6471 Banach-Alaoglu theorem], id=6471 -- WP: [[Banach-Alaoglu theorem]] -- Status: '''A'''

::[[User:AdamSmithee|AdamSmithee]] 15:38, 3 February 2006 (UTC)

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==46B20 Geometry and structure of normed linear spaces==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=5399 \lim_{p \to \infty} \lVert x \rVert_p = \lVert x \rVert_{\infty}], id=5399 -- WP guess: [[\lim_p \to \infty \lVert x \rVert_p = \lVert x \rVert_\infty]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=66115399 Banach-Mazur\lim_{p \to \infty} \lVert x \rVert_p = \lVert x compactum\rVert_{\infty}], id=66115399 -- WP guess: [[Banach-Mazur\lim_p compactum\to \infty \lVert x \rVert_p = \lVert x \rVert_\infty]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=33336611 Banach-Mazur algebracompactum], id=33336611 -- WP guess: [[Banach-Mazur algebracompactum]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6240 basic properties of seminorms], id=6240 -- WP guess: [[basic properties of seminorms]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=3252 Hahn-Banach theorem], id=3252 -- WP guess: [[Hahn-Banach theorem]] -- Status: Status: '''A'''

:: [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 22:58, 10 December 2005 (UTC)

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6515 Minkowski's function], id=6515 -- WP guess: [[Minkowski's function]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4127 proof of Hahn-Banach theorem], id=4127 -- WP guess: [[proof of Hahn-Banach theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=2322 seminorm], id=2322 -- WP guess: [[seminorm]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=91 vector norm], id=91 -- WP guess: [[vector norm]] -- Status:

==46B25 Classical Banach spaces in the general theory==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6270 proof that L^p spaces are complete], id=6270 -- WP: [[Lp space]] -- Status: '''N'''

::[[User:AdamSmithee|AdamSmithee]] 15:54, 3 February 2006 (UTC)

==46B50 Compactness in Banach (or normed) spaces==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4457 proof of Schauder fixed point theorem], id=4457 -- WP guess: [[proof of Schauder fixed point theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=44554457 proof of Schauder fixed point theorem], id=44554457 -- WP guess: [[proof of Schauder fixed point theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4457 proof of4455 Schauder fixed point theorem], id=44574455 -- WP guess: [[proof of Schauder fixed point theorem]] -- Status:

==46B99 Miscellaneous==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=1929 \ell^p], id=1929 -- WP: [[Lp space]] -- Status: '''A'''

:: [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 19:52, 20 November 2005 (UTC)

* PM: [http://planetmath.org/?op=getobj&from=objects&id=5564 all norms on finite-dimensional vector spaces are equivalent], id=5564 -- WP guess: [[all norms on finite-dimensional vector spaces are equivalent]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=1605 Banach space], id=1605 -- WP guess: [[Banach space]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6469 Banach-Steinhaus theorem], id=6469 -- WP: [[Banach-Steinhaus theorem]] -- Status: '''A'''

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::[[User:AdamSmithee|AdamSmithee]] 10:25, 6 February 2006 (UTC)

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4312 equivalent norms], id=4312 -- WP guess: [[equivalent norms]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6633 every finite dimensional normed vector space is a Banach space], id=6633 -- WP guess: [[every finite dimensional normed vector space is a Banach space]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6632 every subspace of a normed space of finite dimension is closed], id=6632 -- WP guess: [[every subspace of a normed space of finite dimension is closed]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=1604 normed vector space], id=1604 -- WP guess: [[normed vector space]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6470 proof of Banach-Steinhaus theorem], id=6470 -- WP: [[Banach-Steinhaus theorem]] -- Status: '''A'''

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==46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=3089 Bessel inequality], id=3089 -- WP: [[Bessel inequality]] -- Status: '''C'''

::[[User:Oleg Alexandrov|Oleg Alexandrov]]

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6364 direct integral of Hilbert spaces], id=6364 -- WP guess: [[direct integral of Hilbert spaces]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6363 direct sum of Hilbert spaces], id=6363 -- WP guess: [[direct sum of Hilbert spaces]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=3401 Hilbert module], id=3401 -- WP guess: [[Hilbert module]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6229 Hilbert parallelotope], id=6229 -- WP guess: [[Hilbert parallelotope]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=1930 Hilbert space], id=1930 -- WP guess: [[Hilbert space]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=3090 proof of Bessel inequality], id=3090 -- WP guess: [[proof of Bessel inequality]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6239 Rellich selection theorem], id=6239 -- WP guess: [[Rellich selection theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=5963 Riesz sequence], id=5963 -- WP guess: [[Riesz sequence]] -- Status:

==46C15 Characterizations of Hilbert spaces==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=1933 classification of separable Hilbert spaces], id=1933 -- WP guess: [[classification of separable Hilbert spaces]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6127 proof of1933 classification of separable Hilbert spaces], id=61271933 -- WP guess: [[proof of classification of separable Hilbert spaces]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=19336127 proof of classification of separable Hilbert spaces], id=19336127 -- WP guess: [[proof of classification of separable Hilbert spaces]] -- Status:

==46C99 Miscellaneous==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=5969 characterization of tight frames in {\mathbb R}^n], id=5969 -- WP guess: [[characterization of tight frames in \mathbb R^n]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=59355969 framecharacterization of tight frames in {\mathbb R}^n], id=59355969 -- WP guess: [[framecharacterization of tight frames in \mathbb R^n]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=59615935 multiresolution analysisframe], id=59615935 -- WP guess: [[multiresolution analysisframe]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=59695961 characterizationmultiresolution of tight frames in {\mathbb R}^nanalysis], id=59695961 -- WP guess: [[characterizationmultiresolution of tight frames in \mathbb R^nanalysis]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6130 proof of Riesz representation theorem for separable Hilbert spaces], id=6130 -- WP guess: [[Riesz representation theorem]] -- Status: '''N'''

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::[[User:AdamSmithee|AdamSmithee]] 16:13, 3 February 2006 (UTC)

* PM: [http://planetmath.org/?op=getobj&from=objects&id=5586 Riesz-Fischer theorem], id=5586 -- WP guess: [[Riesz-Fischer theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=5984 set of sampling], id=5984 -- WP guess: [[set of sampling]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=5959 wavelet], id=5959 -- WP guess: [[wavelet]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=5971 wavelet set], id=5971 -- WP guess: [[wavelet set]] -- Status:

==46E15 Banach spaces of continuous, differentiable or analytic functions==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=2961 Ascoli-Arzelà theorem], id=2961 -- WP guess: [[Ascoli-Arzelà theorem]] -- Status: '''M'''

:: [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 06:57, 25 October 2005 (UTC)

* PM: [http://planetmath.org/?op=getobj&from=objects&id=3753 proof of Ascoli-Arzelà theorem], id=3753 -- WP guess: [[proof of Ascoli-Arzelà theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6146 proof of Stone-Weierstrass theorem], id=6146 -- WP guess: [[proof of Stone-Weierstrass theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=2984 Stone-Weierstrass theorem], id=2984 -- WP guess: [[Stone-Weierstrass theorem]] -- Status:

==46E30 Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=2051 conjugate index], id=2051 -- WP guess: [[conjugate index]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=942051 Holderconjugate inequalityindex], id=942051 -- WP guess: [[Holderconjugate inequalityindex]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4110 proof of94 Holder inequality], id=411094 -- WP guess: [[proof of Holder inequality]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=40794110 proof of YoungHolder Inequalityinequality], id=40794110 -- WP guess: [[proof of YoungHolder Inequalityinequality]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=40784079 proof of Young Inequality], id=40784079 -- WP guess: [[proof of Young Inequality]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=9024078 vectorYoung fieldInequality], id=9024078 -- WP guess: [[vectorYoung fieldInequality]] -- Status:

==46E35 Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6601 Sobolev space], id=6601 -- WP: [[Sobolev space]] -- Status: '''A'''

:[[User:Oleg Alexandrov|Oleg Alexandrov]] 22:50, 13 September 2005 (UTC)

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==46E40 Spaces of vector- and operator-valued functions==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=902 vector field], id=902 -- WP guess: [[vector field]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=2051902 conjugatevector indexfield], id=2051902 -- WP guess: [[conjugatevector indexfield]] -- Status:

==46F05 Topological linear spaces of test functions, distributions and ultradistributions==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=5655 example of Dirac sequence], id=5655 -- WP guess: [[example of Dirac sequence]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=53995655 \lim_{pexample \toof \infty}Dirac \lVert x \rVert_p = \lVert x \rVert_{\infty}sequence], id=53995655 -- WP guess: [[\lim_pexample \toof \inftyDirac \lVert x \rVert_p = \lVert x \rVert_\inftysequence]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4444 space of rapidly decreasing functions], id=4444 -- WP: [[Schwartz space]] -- Status: '''C'''

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==46G05 Derivatives==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=2975 derivative], id=2975 -- WP guess: [[derivative]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=63952975 higher order derivatives of sine and cosinederivative], id=63952975 -- WP guess: [[higher order derivatives of sine and cosinederivative]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6395 higher order derivatives of sine and cosine], id=6395 -- WP guess: [[higher order derivatives of sine and cosine]] -- Status:

==46H05 General theory of topological algebras==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=3333 Banach algebra], id=3333 -- WP guess: [[Banach algebra]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=56553333 exampleBanach of Dirac sequencealgebra], id=56553333 -- WP guess: [[exampleBanach of Dirac sequencealgebra]] -- Status:

==46H35 Topological algebras of operators==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=6402 topological *-algebra], id=6402 -- WP guess: [[topological *-algebra]] -- Status:

==46K05 General theory of topological algebras with involution==

==46L05 General theory of $C^*$-algebras==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=3334 C^*-algebra], id=3334 -- WP: [[C*-algebra]] -- Status: '''A'''

:: [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 23:00, 10 December 2005 (UTC)

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:: Our article contains the relevant bit of info. The proof itself which is contained in the PM article, is not wanted. [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 23:02, 10 December 2005 (UTC)

* PM: [http://planetmath.org/?op=getobj&from=objects&id=3335 Gelfand-Naimark representation theorem], id=3335 -- WP guess: [[Gelfand-Naimark representation theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4574 state], id=4574 -- WP guess: [[state]] -- Status:

==46L85 Noncommutative topology==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4065 Gelfand-Naimark theorem], id=4065 -- WP: [[Gelfand-Naimark theorem]] -- Status: '''A'''

:: Ours is more elementary and more complete. [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 23:03, 10 December 2005 (UTC)

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==46L87 Noncommutative differential geometry==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=3329 Fredholm module], id=3329 -- WP guess: [[Fredholm module]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=29753329 derivativeFredholm module], id=29753329 -- WP guess: [[derivativeFredholm module]] -- Status:

==46M05 Tensor products==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=899 triple scalar product], id=899 -- WP guess: [[triple scalar product]] -- Status:

* PM: [http://planetmath.org/?op=getobj&from=objects&id=3329899 Fredholmtriple modulescalar product], id=3329899 -- WP guess: [[Fredholmtriple modulescalar product]] -- Status:

==46T12 Measure (Gaussian, cylindrical, etc.) and integrals (Feynman, path, Fresnel, etc.) on manifolds==

* PM: [http://planetmath.org/?op=getobj&from=objects&id=1700 path integral], id=1700 -- WP guess: [[path integral]] -- Status: