Home Metamath Proof ExplorerTheorem List (p. 222 of 309) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21328) Hilbert Space Explorer (21329-22851) Users' Mathboxes (22852-30843)

Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempjadjii 22101 A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)

Theorempjaddii 22102 Projection of vector sum is sum of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theorempjinormii 22103 The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)

Theorempjmulii 22104 Projection of (scalar) product is product of projection. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theorempjsubii 22105 Projection of vector difference is difference of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theorempjsslem 22106 Lemma for subset relationships of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theorempjss2i 22107 Subset relationship for projections. Theorem 4.5(i)->(ii) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theorempjssmii 22108 Projection meet property. Remark in [Kalmbach] p. 66. Also Theorem 4.5(i)->(iv) of [Beran] p. 112. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theorempjssge0ii 22109 Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)

Theorempjdifnormii 22110 Theorem 4.5(v)<->(vi) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.)

Theorempjcji 22111 The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)

Theorempjadji 22112 A projection is self-adjoint. Property (i) of [Beran] p. 109. (Contributed by NM, 6-Oct-2000.) (New usage is discouraged.)

Theorempjaddi 22113 Projection of vector sum is sum of projections. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theorempjinormi 22114 The inner product of a projection and its argument is the square of the norm of the projection. Remark in [Halmos] p. 44. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theorempjsubi 22115 Projection of vector difference is difference of projections. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theorempjmuli 22116 Projection of scalar product is scalar product of projection. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjige0i 22117 The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theorempjige0 22118 The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theorempjcjt2 22119 The projection on a subspace join is the sum of the projections. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)

Theorempj0i 22120 The projection of the zero vector. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.)

Theorempjch 22121 Projection of a vector in the projection subspace. Lemma 4.4(ii) of [Beran] p. 111. (Contributed by NM, 30-Oct-1999.) (New usage is discouraged.)

Theorempjid 22122 The projection of a vector in the projection subspace is itself. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)

Theorempjvec 22123* The set of vectors belonging to the subspace of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)

Theorempjocvec 22124* The set of vectors belonging to the orthocomplemented subspace of a projection. Second part of Theorem 27.3 of [Halmos] p. 45. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorempjocini 22125 Membership of projection in orthocomplement of intersection. (Contributed by NM, 21-Apr-2001.) (New usage is discouraged.)

Theorempjini 22126 Membership of projection in an intersection. (Contributed by NM, 22-Apr-2001.) (New usage is discouraged.)

Theorempjjsi 22127* A sufficient condition for subspace join to be equal to subspace sum. (Contributed by NM, 29-May-2004.) (New usage is discouraged.)

Theorempjfni 22128 Functionality of a projection. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theorempjrni 22129 The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

Theorempjfoi 22130 A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorempjfi 22131 The mapping of a projection. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)

Theorempjvi 22132 The value of a projection in terms of components. (Contributed by NM, 28-Nov-2000.) (New usage is discouraged.)

Theorempjhfo 22133 A projection maps onto its subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorempjrn 22134 The range of a projection. Part of Theorem 26.2 of [Halmos] p. 44. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorempjhf 22135 The mapping of a projection. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorempjfn 22136 Functionality of a projection. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)

Theorempjsumi 22137 The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000.) (New usage is discouraged.)

Theorempj11i 22138 One-to-one correspondence of projection and subspace. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theorempjdsi 22139 Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 21-Jun-2006.) (New usage is discouraged.)

Theorempjds3i 22140 Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)

Theorempj11 22141 One-to-one correspondence of projection and subspace. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorempjmfn 22142 Functionality of the projection function. (Contributed by NM, 24-Apr-2006.) (New usage is discouraged.)

Theorempjmf1 22143 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 22144 The inner product of projections on orthogonal subspaces vanishes. (Contributed by NM, 30-Jun-2006.) (New usage is discouraged.)

Theorempjoi0i 22145 The inner product of projections on orthogonal subspaces vanishes. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)

Theorempjopythi 22146 Pythagorean theorem for projections on orthogonal subspaces. (Contributed by NM, 1-Nov-1999.) (New usage is discouraged.)

Theorempjopyth 22147 Pythagorean theorem for projections on orthogonal subspaces. (Contributed by NM, 2-Nov-1999.) (New usage is discouraged.)

Theorempjnormi 22148 The norm of the projection is less than or equal to the norm. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)

Theorempjpythi 22149 Pythagorean theorem for projections. (Contributed by NM, 27-Oct-1999.) (New usage is discouraged.)

Theorempjneli 22150 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 22151 The norm of the projection is less than or equal to the norm. (Contributed by NM, 28-Oct-1999.) (New usage is discouraged.)

Theorempjpyth 22152 Pythagorean theorem for projectors. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)

Theorempjnel 22153 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 22154 A vector belongs to the subspace of a projection iff the norm of its projection equals its norm. This and pjch 22121 yield Theorem 26.3 of [Halmos] p. 44. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.)

15.9.32  Mayet's equation E_3

Theoremmayete3i 22155 Mayet's equation E3. Part of Theorem 4.1 of [Mayet3] p. 1223. (Contributed by NM, 22-Jun-2006.) (New usage is discouraged.)

Theoremmayete3iOLD 22156 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 22157 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.)

15.9.33  Zero and identity operators

Definitiondf-h0op 22158 Define the Hilbert space zero operator. See df0op2 22162 for alternate definition. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Definitiondf-iop 22159 Define the Hilbert space identity operator. See dfiop2 22163 for alternate definition. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)

Theoremho0val 22160 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 22161 Functionality of the zero Hilbert space operator. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theoremdf0op2 22162 Alternate definition of Hilbert space zero operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)

Theoremdfiop2 22163 Alternate definition of Hilbert space identity operator. (Contributed by NM, 7-Aug-2006.) (New usage is discouraged.)

Theoremhoif 22164 Functionality of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)

Theoremhoival 22165 The value of the Hilbert space identity operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)

Theoremhoico1 22166 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhoico2 22167 Composition with the Hilbert space identity operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

15.9.34  Operations on Hilbert space operators

Theoremhoaddcl 22168 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 22169 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 22170* Equality of Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremhoeqi 22171* Equality of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theoremhoscli 22172 Closure of Hilbert space operator sum. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theoremhodcli 22173 Closure of Hilbert space operator difference. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theoremhocoi 22174 Composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theoremhococli 22175 Closure of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theoremhocofi 22176 Mapping of composition of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theoremhocofni 22177 Functionality of composition of Hilbert space operators. (Contributed by NM, 12-Nov-2000.) (New usage is discouraged.)

Theoremhoaddcli 22178 Mapping of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theoremhosubcli 22179 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 22180 Functionality of sum of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.)

Theoremhosubfni 22181 Functionality of difference of Hilbert space operators. (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.)

Theoremhoaddcomi 22182 Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)

Theoremhosubcl 22183 Mapping of difference of Hilbert space operators. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)

Theoremhoaddcom 22184 Commutativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhodsi 22185 Relationship between Hilbert space operator difference and sum. (Contributed by NM, 17-Nov-2000.) (New usage is discouraged.)

Theoremhoaddassi 22186 Associativity of sum of Hilbert space operators. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theoremhoadd12i 22187 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 22188 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 22189 Distributive law for Hilbert space operator sum. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theoremhocsubdiri 22190 Distributive law for Hilbert space operator difference. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.)

Theoremho2coi 22191 Double composition of Hilbert space operators. (Contributed by NM, 1-Dec-2000.) (New usage is discouraged.)

Theoremhoaddass 22192 Associativity of sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhoadd32 22193 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 22194 Rearrangement of 4 terms in a sum of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhocsubdir 22195 Distributive law for Hilbert space operator difference. (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)

Theoremhoaddid1i 22196 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)

Theoremhodidi 22197 Difference of a Hilbert space operator from itself. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theoremho0coi 22198 Composition of the zero operator and a Hilbert space operator. (Contributed by NM, 9-Aug-2006.) (New usage is discouraged.)

Theoremhoid1i 22199 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)

Theoremhoid1ri 22200 Composition of Hilbert space operator with unit identity. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30843
 Copyright terms: Public domain < Previous  Next >