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

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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 (mathematics)|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)

* PM: [http://planetmath.org/?op=getobj&from=objects&id=4473 \operatorname{p.\!v.}(\frac{1}{x}) is a distribution of first order], id=4473 -- WP: [[Distribution (mathematics)|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]]? [[User:Jitse Niesen|Jitse Niesen]] 13:32, 31 Jan 2005 (UTC)

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* 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 (mathematics)|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:

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* PM: [http://planetmath.org/?op=getobj&from=objects&id=6723 weak convergence], id=6723 -- WP guess: [[weak convergence]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8219 Hilbert basis], id=8219^{<font style="color=:red">new!</font>} -- WP guess: [[Hilbert basis]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9771 invariant subspace problem], id=9771^{<font style="color=:red">new!</font>} -- WP guess: [[invariant subspace problem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8208 modular mappings in vector spaces over the field of complex numbers], id=8208^{<font style="color=:red">new!</font>} -- WP guess: [[modular mappings in vector spaces over the field of complex numbers]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8374 modular space], id=8374^{<font style="color=:red">new!</font>} -- WP guess: [[modular space]] -- Status:

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

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* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=6855 weak-* topology], id=6855 -- WP guess: [[weak-* topology]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9703 anti-cone], id=9703^{<font style="color=:red">new!</font>} -- WP guess: [[anti-cone]] -- Status:

==46A08 Barrelled spaces, bornological spaces==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8902 barrel], id=8902^{<font style="color=:red">new!</font>} -- WP guess: [[barrel]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8003 bornological space], id=8003^{<font style="color=:red">new!</font>} -- WP guess: [[bornological space]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8004 bounded set], id=8004^{<font style="color=:red">new!</font>} -- WP guess: [[bounded set]] -- Status:

==46A20 Duality theory==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9667 hyperplane separation], id=9667^{<font style="color=:red">new!</font>} -- WP guess: [[hyperplane separation]] -- Status:

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

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* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=6911 proof of Heine-Cantor theorem], id=6911 -- WP guess: [[proof of Heine-Cantor theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9857 Riesz representation theorem (of linear functionals on function spaces)], id=9857^{<font style="color=:red">new!</font>} -- WP guess: [[Riesz representation theorem (of linear functionals on function spaces)]] -- Status:

==46Axx Topological linear spaces and related structures==

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==46B04 Isometric theory of Banach spaces==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8524 Mazur-Ulam theorem], id=8524^{<font style="color=:red">new!</font>} -- WP guess: [[Mazur-Ulam theorem]] -- Status:

==46B07 Local theory of Banach spaces==

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* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=6967 sub-linear], id=6967 -- WP guess: [[sub-linear]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9087 Golab's theorem], id=9087^{<font style="color=:red">new!</font>} -- WP guess: [[Golab's theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9086 normed plane], id=9086^{<font style="color=:red">new!</font>} -- WP guess: [[normed plane]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8823 nuclear space], id=8823^{<font style="color=:red">new!</font>} -- WP guess: [[nuclear space]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=7704 properties of Minkowski's functional], id=7704^{<font style="color=:red">new!</font>} -- WP guess: [[properties of Minkowski's functional]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=92 vector p-norm], id=92^{<font style="color=:red">new!</font>} -- WP guess: [[vector p-norm]] -- Status:

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

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* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=4455 Schauder fixed point theorem], id=4455 -- WP guess: [[Schauder fixed point theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8125 Tychonoff fixed point theorem], id=8125^{<font style="color=:red">new!</font>} -- WP guess: [[Tychonoff fixed point theorem]] -- Status:

==46B99 Miscellaneous==

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* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=7092 shift operators in \ell^p], id=7092 -- WP guess: [[shift operators in \ell^p]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9864 approximation property], id=9864^{<font style="color=:red">new!</font>} -- WP guess: [[approximation property]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9751 necessary and sufficient conditions for a normed vector space to be a Banach space], id=9751^{<font style="color=:red">new!</font>} -- WP guess: [[necessary and sufficient conditions for a normed vector space to be a Banach space]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9749 quotient norm], id=9749^{<font style="color=:red">new!</font>} -- WP guess: [[quotient norm]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9750 quotients of Banach spaces by closed subspaces are Banach spaces under the quotient norm], id=9750^{<font style="color=:red">new!</font>} -- WP guess: [[quotients of Banach spaces by closed subspaces are Banach spaces under the quotient norm]] -- Status:

==46Bxx Normed linear spaces and Banach spaces; Banach lattices==

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* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=7271 \vert\langle Tv,v \rangle\vert \leq {\mu} \Vert v \Vert ^2 for all v implies \Vert T \Vert \leq {\mu}], id=7271 -- WP guess: [[\vert\langle Tv,v \rangle\vert \leq \mu \Vert v \Vert ^2 for all v implies \Vert T \Vert \leq \mu]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=7691 example of non-separable Hilbert space], id=7691^{<font style="color=:red">new!</font>} -- WP guess: [[example of non-separable Hilbert space]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8227 generalization of the parallelogram law], id=8227^{<font style="color=:red">new!</font>} -- WP guess: [[generalization of the parallelogram law]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8228 Hlawka's inequality], id=8228^{<font style="color=:red">new!</font>} -- WP guess: [[Hlawka's inequality]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8205 parallelogram law], id=8205^{<font style="color=:red">new!</font>} -- WP guess: [[parallelogram law]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8229 proof of generalization of the parallelogram law], id=8229^{<font style="color=:red">new!</font>} -- WP guess: [[proof of generalization of the parallelogram law]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8211 proof of parallelogram law], id=8211^{<font style="color=:red">new!</font>} -- WP guess: [[proof of parallelogram law]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=7967 Schur's condition for a matrix to be a bounded operator on l^2], id=7967^{<font style="color=:red">new!</font>} -- WP guess: [[Schur's condition for a matrix to be a bounded operator on l^2]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8214 yet another proof of parallelogram law], id=8214^{<font colorstyle="color:red">new!</fontsup><sup style="color:red">new!} -- WP guess: [[yet another proof of parallelogram law]] -- Status:

==46C15 Characterizations of Hilbert spaces==

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

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8100 Banach spaces with complemented subspaces], id=8100^{<font style="color=:red">new!</font>} -- WP guess: [[Banach spaces with complemented subspaces]] -- Status:

==46C99 Miscellaneous==

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

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9448 Gabor frame], id=9448^{<font style="color=:red">new!</font>} -- WP guess: [[Gabor frame]] -- Status:

==46Cxx Inner product spaces and their generalizations, Hilbert spaces==

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:: [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 04:10, 11 March 2006 (UTC)

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=7667 generalization of Young inequality], id=7667^{<font style="color=:red">new!</font>} -- WP guess: [[generalization of Young inequality]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9170 generalized Hölder inequality], id=9170^{<font style="color=:red">new!</font>} -- WP guess: [[generalized Hölder inequality]] -- Status:

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

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

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

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8221 Fréchet derivative is unique], id=8221^{<font style="color=:red">new!</font>} -- WP guess: [[Fréchet derivative is unique]] -- Status:

==46Gxx Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces)==

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* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=7028 uniformly convex space is reflexive], id=7028 -- WP guess: [[uniformly convex space is reflexive]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9743 Gelfand transform], id=9743^{<font style="color=:red">new!</font>} -- WP guess: [[Gelfand transform]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9871 Gelfand-Mazur theorem], id=9871^{<font colorstyle="color:red">new!</fontsup><sup style="color:red">new!} -- WP guess: [[Gelfand-Mazur theorem]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9757 invertible elements in a Banach algebra form an open set], id=9757^{<font style="color=:red">new!</font>} -- WP guess: [[invertible elements in a Banach algebra form an open set]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9737 multiplicative linear functional], id=9737^{<font style="color=:red">new!</font>} -- WP guess: [[multiplicative linear functional]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9756 Neumann series in Banach algebras], id=9756^{<font style="color=:red">new!</font>} -- WP guess: [[Neumann series in Banach algebras]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8286 normed algebra], id=8286^{<font style="color=:red">new!</font>} -- WP guess: [[normed algebra]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8141 rotund space], id=8141^{<font style="color=:red">new!</font>} -- WP guess: [[rotund space]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9791 spectrum is a non-empty compact set], id=9791^{<font style="color=:red">new!</font>} -- WP guess: [[spectrum is a non-empty compact set]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8299 topological divisor of zero], id=8299^{<font style="color=:red">new!</font>} -- WP guess: [[topological divisor of zero]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=8295 topologically nilpotent], id=8295^{<font style="color=:red">new!</font>} -- WP guess: [[topologically nilpotent]] -- Status:

==46H15 Representations of topological algebras==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9843 Banach *-algebra representation], id=9843^{<font style="color=:red">new!</font>} -- WP guess: [[Banach *-algebra representation]] -- Status:

==46H35 Topological algebras of operators==

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==46K10 Representations of topological algebras with involution==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9845 criterion for a Banach *-algebra representation to be irreducible], id=9845^{<font style="color=:red">new!</font>} -- WP guess: [[criterion for a Banach *-algebra representation to be irreducible]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9844 representations of Banach *-algebras are continuous], id=9844^{<font style="color=:red">new!</font>} -- WP guess: [[representations of Banach *-algebras are continuous]] -- Status:

==46Kxx Topological (rings and) algebras with an involution==

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

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9807 example of Banach algebra which is not a C^*-algebra for any involution], id=9807^{<font style="color=:red">new!</font>} -- WP guess: [[example of Banach algebra which is not a C^*-algebra for any involution]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9862 special elements in a C^*-algebra and their spectral properties], id=9862^{<font colorstyle="color:red">new!</fontsup><sup style="color:red">new!} -- WP guess: [[special elements in a C^*-algebra and their spectral properties]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9846 topologically irreducible representations are algebrically irreducible for C^*-algebras], id=9846^{<font style="color=:red">new!</font>} -- WP guess: [[topologically irreducible representations are algebrically irreducible for C^*-algebras]] -- Status:

==46L10 General theory of von Neumann algebras==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9868 polar decomposition in von Neumann algebras], id=9868^{<font style="color=:red">new!</font>} -- WP guess: [[polar decomposition in von Neumann algebras]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9722 von Neumann algebra], id=9722^{<font style="color=:red">new!</font>} -- WP guess: [[von Neumann algebra]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9869 von Neumann algebras contain the range projections of its elements], id=9869^{<font style="color=:red">new!</font>} -- WP guess: [[von Neumann algebras contain the range projections of its elements]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9870 von Neumann algebras of dimension greater than one contain non-trivial projections], id=9870^{<font style="color=:red">new!</font>} -- WP guess: [[von Neumann algebras of dimension greater than one contain non-trivial projections]] -- Status:

==46L65 Quantizations, deformations==

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* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=7540 Quantization], id=7540 -- WP guess: [[Quantization]] -- Status:

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=7894 canonical quantization], id=7894^{<font style="color=:red">new!</font>} -- WP guess: [[canonical quantization]] -- Status:

==46L85 Noncommutative topology==

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==46N99 Miscellaneous==

* PM: [http://planetmath.org/?op=getobj&amp;from=objects&amp;id=9629 Schwarzian derivative], id=9629^{<font style="color=:red">new!</font>} -- WP guess: [[Schwarzian derivative]] -- Status:

==46Nxx Miscellaneous applications of functional analysis==