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Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempjfn 23201 Functionality of a projection. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)

Theorempjsumi 23202 The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)

Theorempj11i 23203 One-to-one correspondence of projection and subspace. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjdsi 23204 Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 21-Jun-2006.) (New usage is discouraged.)

Theorempjds3i 23205 Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)

Theorempj11 23206 One-to-one correspondence of projection and subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorempjmfn 23207 Functionality of the projection function. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorempjmf1 23208 The projector function maps one-to-one into the set of Hilbert space operators. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorempjoi0 23209 The inner product of projections on orthogonal subspaces vanishes. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theorempjoi0i 23210 The inner product of projections on orthogonal subspaces vanishes. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)

Theorempjopythi 23211 Pythagorean theorem for projections on orthogonal subspaces. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)

Theorempjopyth 23212 Pythagorean theorem for projections on orthogonal subspaces. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theorempjnormi 23213 The norm of the projection is less than or equal to the norm. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)

Theorempjpythi 23214 Pythagorean theorem for projections. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)

Theorempjneli 23215 If a vector does not belong to subspace, the norm of its projection is less than its norm. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)

Theorempjnorm 23216 The norm of the projection is less than or equal to the norm. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)

Theorempjpyth 23217 Pythagorean theorem for projectors. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)

Theorempjnel 23218 If a vector does not belong to subspace, the norm of its projection is less than its norm. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theorempjnorm2 23219 A vector belongs to the subspace of a projection iff the norm of its projection equals its norm. This and pjch 23186 yield Theorem 26.3 of [Halmos] p. 44. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)

18.5.11  Mayet's equation E_3

Theoremmayete3i 23220 Mayet's equation E3. Part of Theorem 4.1 of [Mayet3] p. 1223. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)

Theoremmayete3iOLD 23221 Mayet's equation E3. Part of Theorem 4.1 of [Mayet3] p. 7. (Contributed by NM, 22-Jun-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremmayetes3i 23222 Mayet's equation E^*3, derived from E3. Solution, for n = 3, to open problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. (Contributed by NM, 10-May-2009.) (New usage is discouraged.)

18.6  Operators on Hilbert spaces

18.6.1  Operator sum, difference, and scalar multiplication

Note on operators. Unlike some authors, we use the term "operator" to mean any function from to . This is the definition of operator in [Hughes] p. 14, the definition of operator in [AkhiezerGlazman] p. 30, and the definition of operator in [Goldberg] p. 10. For Reed and Simon, an operator is linear (definition of operator in [ReedSimon] p. 2). For Halmos, an operator is bounded and linear (definition of operator in [Halmos] p. 35). For Kalmbach and Beran, an operator is continuous and linear (definition of operator in [Kalmbach] p. 353; definition of operator in [Beran] p. 99). Note that "bounded and linear" and "continuous and linear" are equivalent by lncnbd 23531.

Definitiondf-hosum 23223* Define the sum of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)

Definitiondf-homul 23224* Define the scalar product with a Hilbert space operator. Definition of [Beran] p. 111. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Definitiondf-hodif 23225* Define the difference of two Hilbert space operators. Definition of [Beran] p. 111. (Contributed by NM, 9-Nov-2000.) (New usage is discouraged.)

Definitiondf-hfsum 23226* Define the sum of two Hilbert space functionals. Definition of [Beran] p. 111. Note that unlike some authors, we define a functional as any function from to , not just linear (or bounded linear) ones. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Definitiondf-hfmul 23227* Define the scalar product with a Hilbert space functional. Definition of [Beran] p. 111. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Theoremhosmval 23228* Value of the sum of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Theoremhommval 23229* Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Theoremhodmval 23230* Value of the difference of two Hilbert space operators. (Contributed by NM, 9-Nov-2000.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Theoremhfsmval 23231* Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Theoremhfmmval 23232* Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Theoremhosval 23233 Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhomval 23234 Value of the scalar product with a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhodval 23235 Value of the difference of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhfsval 23236 Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhfmval 23237 Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhoscl 23238 Closure of the sum of two Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theoremhomcl 23239 Closure of the scalar product of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremhodcl 23240 Closure of the difference of two Hilbert space operators. (Contributed by NM, 15-Nov-2002.) (New usage is discouraged.)

18.6.2  Zero and identity operators

Definitiondf-h0op 23241 Define the Hilbert space zero operator. See df0op2 23245 for alternate definition. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Definitiondf-iop 23242 Define the Hilbert space identity operator. See dfiop2 23246 for alternate definition. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)

Theoremho0val 23243 Value of the zero Hilbert space operator (null projector). Remark in [Beran] p. 111. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Theoremho0f 23244 Functionality of the zero Hilbert space operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theoremdf0op2 23245 Alternate definition of Hilbert space zero operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)

Theoremdfiop2 23246 Alternate definition of Hilbert space identity operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)

Theoremhoif 23247 Functionality of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)

Theoremhoival 23248 The value of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)

Theoremhoico1 23249 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhoico2 23250 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

18.6.3  Operations on Hilbert space operators

Theoremhoaddcl 23251 The sum of Hilbert space operators is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhomulcl 23252 The scalar product of a Hilbert space operator is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhoeq 23253* Equality of Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremhoeqi 23254* Equality of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theoremhoscli 23255 Closure of Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theoremhodcli 23256 Closure of Hilbert space operator difference. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theoremhocoi 23257 Composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theoremhococli 23258 Closure of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theoremhocofi 23259 Mapping of composition of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theoremhocofni 23260 Functionality of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theoremhoaddcli 23261 Mapping of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theoremhosubcli 23262 Mapping of difference of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhoaddfni 23263 Functionality of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theoremhosubfni 23264 Functionality of difference of Hilbert space operators. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theoremhoaddcomi 23265 Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)

Theoremhosubcl 23266 Mapping of difference of Hilbert space operators. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)

Theoremhoaddcom 23267 Commutativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhodsi 23268 Relationship between Hilbert space operator difference and sum. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)

Theoremhoaddassi 23269 Associativity of sum of Hilbert space operators. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theoremhoadd12i 23270 Commutative/associative law for Hilbert space operator sum that swaps the first two terms. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)

Theoremhoadd32i 23271 Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)

Theoremhocadddiri 23272 Distributive law for Hilbert space operator sum. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theoremhocsubdiri 23273 Distributive law for Hilbert space operator difference. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theoremho2coi 23274 Double composition of Hilbert space operators. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)

Theoremhoaddass 23275 Associativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhoadd32 23276 Commutative/associative law for Hilbert space operator sum that swaps the second and third terms. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhoadd4 23277 Rearrangement of 4 terms in a sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhocsubdir 23278 Distributive law for Hilbert space operator difference. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)

Theoremhoaddid1i 23279 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)

Theoremhodidi 23280 Difference of a Hilbert space operator from itself. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theoremho0coi 23281 Composition of the zero operator and a Hilbert space operator. (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)

Theoremhoid1i 23282 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)

Theoremhoid1ri 23283 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)

Theoremhoaddid1 23284 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)

Theoremhodid 23285 Difference of a Hilbert space operator from itself. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Theoremhon0 23286 A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theoremhodseqi 23287 Subtraction and addition of equal Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)

Theoremho0subi 23288 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theoremhonegsubi 23289 Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremho0sub 23290 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)

Theoremhosubid1 23291 The zero operator subtracted from a Hilbert space operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)

Theoremhonegsub 23292 Relationship between Hilbert space operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhomulid2 23293 An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremhomco1 23294 Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)

Theoremhomulass 23295 Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremhoadddi 23296 Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremhoadddir 23297 Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)

Theoremhomul12 23298 Swap first and second factors in a nested operator scalar product. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremhonegneg 23299 Double negative of a Hilbert space operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhosubneg 23300 Relationship between operator subtraction and negative. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)

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