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

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This page provides a list of all articles available at PlanetMath in the following topic:

46-XX Functional analysis.

This list will be periodically updated. Each entry in the list has three fields:

1. PM : The first field is the link to the PlanetMath article, along with the article's object ID.
2. WP : The second field is either a "guessed" link to a correspondingly named Wikipedia article, produced by the script which generated the list, or one or more manually entered links to the corresponding Wikipedia articles on the subject.
3. Status : The third field is the status field, which explains the current status of the entry. The recommended status entries are:

Status	means PM article
N	not needed
A	adequately covered
C	copied
M	merged
NC	needs copying
NM	needs merging

• Please update the WP and Status fields as appropriate.
• if the WP field is correct please remove the qualifier "guess".
• If the corresponding Wikipedia article exists, but the link to it is wrong, please fix the link.
• If you copy or merge an article from PlanetMath, please update the WP and Status fields for that entry.
• If you have any comments, for example, thoughts on how the PlanetMath article compares to the corresponding Wikipedia article(s), please place such comments on a new indented line following the entry. Comments of this kind are very valuable.

Don't forget to include the relevant template if you copy over text or feel like an external link is warranted

• {{planetmath|id=|title=}} for copied over text
• {{planetmath reference|id=|title=}} for an external link

See the main page for examples and usage criteria.

One can use the web-based program Pmform to convert PlanetMath articles to the Wikipedia format. As a side benefit, this tool will place the PlanetMath template for you.

[edit] 46-00 General reference works (handbooks, dictionaries, bibliographies, etc.)

• PM: 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. Jitse Niesen 13:32, 31 Jan 2005 (UTC)

• PM: \operatorname{p.\!v.}(\frac{1}{x}) is a distribution of first order, id=4473 -- WP: distribution -- Status:

PM article is a proof which is not included in WP; p.v. (1/x) is a nice example of a distribution which should be mentioned in WP. Connection with renormalization? Jitse Niesen 13:32, 31 Jan 2005 (UTC)

• PM: balanced set, id=4425 -- WP guess: balanced set -- Status: NC

No WP article, could perhaps be included in Fréchet space. Jitse Niesen 13:32, 31 Jan 2005 (UTC)

• PM: bounded function, id=4426 -- WP: bounded, bounded function, Banach space -- Status: NM

PM definition is included in bounded, while bounded function uses a different definition coming from order theory. The space of bounded functions is treated as an example in Banach space. Jitse Niesen 13:32, 31 Jan 2005 (UTC)

• PM: bounded set (in a topological vector space), id=4429 -- WP: bounded set, bounded set (topological vector space) -- Status: A

PM definition is more general. Need to create article bounded set, possibly as a redirect to bounded. The thm in PM (compact implies bounded) should be added. Jitse Niesen 13:32, 31 Jan 2005 (UTC)

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

• PM: Cauchy principal part integral, id=4472 -- WP: Cauchy principal value -- Status: M

Copied the most essential part, as to not disturb too much the existing article. Oleg Alexandrov (talk) 04:52, 10 March 2006 (UTC)

• PM: Cauchy sequence, id=6195 -- WP: Cauchy sequence -- Status: A

Our artilcle is more complete. Paul August ☎ 15:03, Jan 31, 2005 (UTC)

• PM: cone, id=4570 -- WP guess: cone -- Status:

• PM: delta distribution, id=4468 -- WP guess: delta distribution -- Status:

• PM: distribution, id=4427 -- WP guess: distribution -- Status:

• PM: every locally integrable function is a distribution, id=4433 -- WP guess: every locally integrable function is a distribution -- Status:

• PM: function spaces, id=5556 -- WP guess: function spaces -- Status:

• PM: localization for distributions, id=4477 -- WP guess: localization for distributions -- Status:

• PM: locally convex topological vector space, id=4424 -- WP guess: locally convex topological vector space -- Status:

• PM: operations on distributions, id=4440 -- WP guess: operations on distributions -- Status:

• PM: proof of convergence theorem, id=4437 -- WP guess: proof of convergence theorem -- Status:

• PM: sequential characterization of boundedness, id=4525 -- WP guess: sequential characterization of boundedness -- Status:

• PM: smooth distribution, id=4568 -- WP guess: smooth distribution -- Status:

• PM: support of distribution, id=4484 -- WP guess: support of distribution -- Status:

• PM: symmetric set, id=4528 -- WP guess: symmetric set -- Status:

• PM: balanced set, id=7453^new! -- WP guess: balanced set -- Status:

• PM: boundedness in a topological vector space generalizes boundedness in a metric space, id=7459^new! -- WP guess: boundedness in a topological vector space generalizes boundedness in a metric space -- Status:

• PM: cone, id=7447^new! -- WP guess: cone -- Status:

• PM: weak convergence, id=6723^new! -- WP guess: weak convergence -- Status:

[edit] 46A03 General theory of locally convex spaces

• PM: Krein-Milman theorem, id=5921 -- WP: Krein-Milman theorem -- Status: C

Oleg Alexandrov (talk) 19:16, 20 November 2005 (UTC)

• PM: proof of Krein-Milman theorem, id=7251^new! -- WP guess: proof of Krein-Milman theorem -- Status:

• PM: weak-* topology, id=6855^new! -- WP guess: weak-* topology -- Status:

[edit] 46A08 Barrelled spaces, bornological spaces

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

• PM: closed graph theorem, id=3702 -- WP: closed graph theorem -- Status: A

Oleg Alexandrov (talk) 19:01, 20 November 2005 (UTC)

• PM: open mapping theorem, id=3675 -- WP: open mapping theorem -- Status: A

Oleg Alexandrov (talk) 19:01, 20 November 2005 (UTC)

• PM: proof of closed graph theorem, id=6472 -- WP: proof of closed graph theorem -- Status: N

Oleg Alexandrov (talk) 19:01, 20 November 2005 (UTC)

[edit] 46A55 Convex sets in topological linear spaces; Choquet theory

• PM: 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: proof that the convex hull of S is open if S is open, id=5587 -- WP guess: proof that the convex hull of S is open if S is open -- Status:

• PM: conical neighborhood, id=7137^new! -- WP guess: conical neighborhood -- Status:

[edit] 46A99 Miscellaneous

• PM: Heine-Cantor theorem, id=3066 -- WP: Heine-Cantor theorem -- Status: M

AdamSmithee 15:29, 3 February 2006 (UTC)

• PM: proof of Heine-Cantor theorem, id=4114 -- WP: Heine-Cantor theorem -- Status: N

AdamSmithee 15:29, 3 February 2006 (UTC)

• PM: topological vector space, id=1744 -- WP guess: topological vector space -- Status:

• PM: proof of Heine-Cantor theorem, id=6911^new! -- WP guess: proof of Heine-Cantor theorem -- Status:

[edit] 46B03 Isomorphic theory (including renorming) of Banach spaces

• PM: hyperbolic isomorphism, id=4315 -- WP guess: hyperbolic isomorphism -- Status:

[edit] 46B07 Local theory of Banach spaces

• PM: Lipschitz inverse mapping theorem, id=5927 -- WP guess: Lipschitz inverse mapping theorem -- Status:

[edit] 46B10 Duality and reflexivity

• PM: Banach-Alaoglu theorem, id=6471 -- WP: Banach-Alaoglu theorem -- Status: A

AdamSmithee 15:38, 3 February 2006 (UTC)

• PM: weak* convergence in normed linear space, id=5229 -- WP guess: weak* convergence in normed linear space -- Status: NC

AdamSmithee 15:38, 3 February 2006 (UTC)

• PM: Proof of Banach-Alaoglu theorem, id=6917^new! -- WP guess: Proof of Banach-Alaoglu theorem -- Status:

[edit] 46B20 Geometry and structure of normed linear spaces

• PM: \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: Banach-Mazur compactum, id=6611 -- WP guess: Banach-Mazur compactum -- Status:

• PM: basic properties of seminorms, id=6240 -- WP guess: basic properties of seminorms -- Status:

• PM: Hahn-Banach theorem, id=3252 -- WP guess: Hahn-Banach theorem -- Status: Status: A

Oleg Alexandrov (talk) 22:58, 10 December 2005 (UTC)

• PM: Minkowski's function, id=6515 -- WP guess: Minkowski's function -- Status: A

Oleg Alexandrov (talk) 05:03, 10 March 2006 (UTC)

• PM: proof of Hahn-Banach theorem, id=4127 -- WP guess: proof of Hahn-Banach theorem -- Status:

• PM: seminorm, id=2322 -- WP guess: seminorm -- Status:

• PM: vector norm, id=91 -- WP guess: vector norm -- Status:

• PM: sub-linear, id=6967^new! -- WP guess: sub-linear -- Status:

[edit] 46B25 Classical Banach spaces in the general theory

• PM: proof that L^p spaces are complete, id=6270 -- WP: Lp space -- Status: N

AdamSmithee 15:54, 3 February 2006 (UTC)

[edit] 46B50 Compactness in Banach (or normed) spaces

• PM: proof of Schauder fixed point theorem, id=4457 -- WP guess: proof of Schauder fixed point theorem -- Status:

• PM: Schauder fixed point theorem, id=4455 -- WP guess: Schauder fixed point theorem -- Status:

[edit] 46B99 Miscellaneous

• PM: \ell^p, id=1929 -- WP: Lp space -- Status: A

Oleg Alexandrov (talk) 19:52, 20 November 2005 (UTC)

• PM: all norms on finite-dimensional vector spaces are equivalent, id=5564 -- WP guess: all norms on finite-dimensional vector spaces are equivalent -- Status:

• PM: Banach space, id=1605 -- WP guess: Banach space -- Status:

• PM: Banach-Steinhaus theorem, id=6469 -- WP: Banach-Steinhaus theorem -- Status: A

AdamSmithee 10:25, 6 February 2006 (UTC)

• PM: bounded operator, id=5226 -- WP: bounded operator -- Status: A

Oleg Alexandrov (talk) 06:52, 25 October 2005 (UTC)

• PM: compact operator, id=5966 -- WP: compact operator -- Status: A

Oleg Alexandrov 22:47, 13 September 2005 (UTC)

• PM: continuous linear mapping, id=3741 -- WP: continuous linear mapping and bounded operator -- Status: A

AdamSmithee 10:25, 6 February 2006 (UTC)

• PM: equivalent norms, id=4312 -- WP guess: equivalent norms -- Status:

• PM: 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: 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: normed vector space, id=1604 -- WP guess: normed vector space -- Status:

• PM: proof of Banach-Steinhaus theorem, id=6470 -- WP: Banach-Steinhaus theorem -- Status: A

AdamSmithee 10:25, 6 February 2006 (UTC)

• PM: all norms are not equivalent, id=7519^new! -- WP guess: all norms are not equivalent -- Status:

• PM: extended norm, id=6703^new! -- WP guess: extended norm -- Status:

• PM: scaling of the open ball in a normed vector space, id=7458^new! -- WP guess: scaling of the open ball in a normed vector space -- Status:

• PM: shift operators in \ell^p, id=7092^new! -- WP guess: shift operators in \ell^p -- Status:

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

• PM: Bessel inequality, id=3089 -- WP: Bessel inequality -- Status: C

Oleg Alexandrov 02:07, 5 March 2006 (UTC)

• PM: direct integral of Hilbert spaces, id=6364 -- WP guess: direct integral of Hilbert spaces -- Status:

• PM: direct sum of Hilbert spaces, id=6363 -- WP guess: direct sum of Hilbert spaces -- Status:

• PM: Hilbert module, id=3401 -- WP guess: Hilbert module -- Status:

• PM: Hilbert parallelotope, id=6229 -- WP guess: Hilbert parallelotope -- Status:

• PM: Hilbert space, id=1930 -- WP guess: Hilbert space -- Status:

• PM: proof of Bessel inequality, id=3090 -- WP guess: proof of Bessel inequality -- Status:

• PM: Rellich selection theorem, id=6239 -- WP guess: Rellich selection theorem -- Status:

• PM: Riesz sequence, id=5963 -- WP: Riesz sequence -- Status: C

Along with Riesz basis. Ben Cairns 12:55, 18 April 2006 (UTC)

• PM: \vert\langle Tv,v \rangle\vert \leq {\mu} \Vert v \Vert ^2 for all v implies \Vert T \Vert \leq {\mu}, id=7271^new! -- WP guess: \vert\langle Tv,v \rangle\vert \leq \mu \Vert v \Vert ^2 for all v implies \Vert T \Vert \leq \mu -- Status:

[edit] 46C15 Characterizations of Hilbert spaces

• PM: classification of separable Hilbert spaces, id=1933 -- WP guess: classification of separable Hilbert spaces -- Status:

• PM: proof of classification of separable Hilbert spaces, id=6127 -- WP guess: proof of classification of separable Hilbert spaces -- Status:

[edit] 46C99 Miscellaneous

• PM: characterization of tight frames in {\mathbb R}^n, id=5969 -- WP guess: characterization of tight frames in \mathbb R^n -- Status:

• PM: frame, id=5935 -- WP guess: frame -- Status:

• PM: multiresolution analysis, id=5961 -- WP guess: multiresolution analysis -- Status:

• PM: proof of Riesz representation theorem for separable Hilbert spaces, id=6130 -- WP guess: Riesz representation theorem -- Status: N

AdamSmithee 16:13, 3 February 2006 (UTC)

• PM: Riesz representation theorem, id=5585 -- WP: Riesz representation theorem -- Status: A

AdamSmithee 16:13, 3 February 2006 (UTC)

• PM: Riesz-Fischer theorem, id=5586 -- WP guess: Riesz-Fischer theorem -- Status:

• PM: set of sampling, id=5984 -- WP guess: set of sampling -- Status:

• PM: wavelet, id=5959 -- WP guess: wavelet -- Status:

• PM: wavelet set, id=5971 -- WP guess: wavelet set -- Status:

[edit] 46E15 Banach spaces of continuous, differentiable or analytic functions

• PM: Ascoli-Arzelà theorem, id=2961 -- WP guess: Ascoli-Arzelà theorem -- Status: M

Oleg Alexandrov (talk) 06:57, 25 October 2005 (UTC)

• PM: proof of Ascoli-Arzelà theorem, id=3753 -- WP guess: proof of Ascoli-Arzelà theorem -- Status:

• PM: proof of Stone-Weierstrass theorem, id=6146 -- WP guess: proof of Stone-Weierstrass theorem -- Status:

• PM: Stone-Weierstrass theorem, id=2984 -- WP guess: Stone-Weierstrass theorem -- Status:

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

• PM: conjugate index, id=2051 -- WP: conjugate index -- Status: C

Oleg Alexandrov (talk) 04:43, 11 March 2006 (UTC)

• PM: Holder inequality, id=94 -- WP guess: Holder inequality -- Status:

• PM: proof of Holder inequality, id=4110 -- WP guess: proof of Holder inequality -- Status:

• PM: proof of Young Inequality, id=4079 -- WP guess: proof of Young Inequality -- Status:

• PM: Young Inequality, id=4078 -- WP: Young's inequality -- Status: A

Oleg Alexandrov (talk) 04:10, 11 March 2006 (UTC)

[edit] 46E35 Sobolev spaces and other spaces of ``smooth functions, embedding theorems, trace theorems

• PM: Sobolev space, id=6601 -- WP: Sobolev space -- Status: A

Oleg Alexandrov 22:50, 13 September 2005 (UTC)

• PM: weak derivative, id=6600 -- WP: weak derivative -- Status: A

Oleg Alexandrov 22:50, 13 September 2005 (UTC)

• PM: proof of Sobolev inequality for \Omega=\mathbf{R}^n, id=6816^new! -- WP guess: proof of Sobolev inequality for \Omega=\mathbfR^n -- Status:

• PM: Sobolev inequality, id=6812^new! -- WP guess: Sobolev inequality -- Status:

[edit] 46E40 Spaces of vector- and operator-valued functions

• PM: vector field, id=902 -- WP guess: vector field -- Status:

[edit] 46F05 Topological linear spaces of test functions, distributions and ultradistributions

• PM: example of Dirac sequence, id=5655 -- WP guess: example of Dirac sequence -- Status:

• PM: space of rapidly decreasing functions, id=4444 -- WP: Schwartz space -- Status: C

linas 21:44, 4 March 2006 (UTC)

[edit] 46G05 Derivatives

• PM: derivative, id=2975 -- WP guess: derivative -- Status:

• PM: higher order derivatives of sine and cosine, id=6395 -- WP guess: higher order derivatives of sine and cosine -- Status:

[edit] 46H05 General theory of topological algebras

• PM: Banach algebra, id=3333 -- WP guess: Banach algebra -- Status:

• PM: Hilbert spaces are uniformly convex spaces, id=7021^new! -- WP guess: Hilbert spaces are uniformly convex spaces -- Status:

• PM: Banach-Krein-mulian theorem, id=7026^new! -- WP guess: Banach-Krein-Mulian theorem -- Status:

• PM: Hahn-Banach theorem(geometric form), id=7034^new! -- WP guess: Hahn-Banach theorem (geometric form) -- Status:

• PM: Hyperplane, id=7035^new! -- WP guess: Hyperplane -- Status:

• PM: proof of Hilbert space is uniformly convex space, id=7167^new! -- WP guess: proof of Hilbert space is uniformly convex space -- Status:

• PM: Property of uniformly convex Banach Space, id=7011^new! -- WP guess: Property of uniformly convex Banach Space -- Status:

• PM: Uniformly convex Banach space, id=6983^new! -- WP guess: Uniformly convex Banach space -- Status:

• PM: uniformly convex space is reflexive, id=7028^new! -- WP guess: uniformly convex space is reflexive -- Status:

[edit] 46H35 Topological algebras of operators

• PM: topological *-algebra, id=6402 -- WP guess: topological *-algebra -- Status:

[edit] 46K05 General theory of topological algebras with involution

[edit] 46L05 General theory of $C^*$-algebras

• PM: C^*-algebra, id=3334 -- WP: C*-algebra -- Status: A

Oleg Alexandrov (talk) 23:00, 10 December 2005 (UTC)

• PM: bounded operators on a Hilbert space form a C^*-algebra, id=6438 -- WP: C*-algebra -- Status: A

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

• PM: Gelfand-Naimark representation theorem, id=3335 -- WP guess: Gelfand-Naimark representation theorem -- Status:

• PM: state, id=4574 -- WP guess: state -- Status:

[edit] 46L65 Quantizations, deformations

• PM: Quantization, id=7540^new! -- WP guess: Quantization -- Status:

[edit] 46L85 Noncommutative topology

• PM: Gelfand-Naimark theorem, id=4065 -- WP: Gelfand-Naimark theorem -- Status: A

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

• PM: Serre-Swan theorem, id=4066 -- WP guess: Serre-Swan theorem -- Status:

[edit] 46L87 Noncommutative differential geometry

• PM: Fredholm module, id=3329 -- WP guess: Fredholm module -- Status:

[edit] 46M05 Tensor products

• PM: triple scalar product, id=899 -- WP guess: triple scalar product -- Status:

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

• PM: path integral, id=1700 -- WP guess: path integral -- Status:

[edit] 46T20 Continuous and differentiable maps