Home Metamath Proof ExplorerTheorem List (p. 187 of 328) < Previous  Next > Browser slow? Try the Unicode version.

 Color key: Metamath Proof Explorer (1-21514) Hilbert Space Explorer (21515-23037) Users' Mathboxes (23038-32776)

Theorem List for Metamath Proof Explorer - 18601-18700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremclmvsass 18601 Associative law for scalar product. (lmodvsass 15670 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                     CMod

Theoremclmvsdir 18602 Distributive law for scalar product. (lmodvsdir 15668 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                            CMod

Theoremclmvs1 18603 Scalar product with ring unit. (lmodvs1 15674 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremclm0vs 18604 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 15679 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                     CMod

Theoremclmvneg1 18605 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 15683 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar              CMod

Theoremclmvsneg 18606 Multiplication of a vector by a negated scalar. (lmodvsneg 15685 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                            CMod

Theoremclmmulg 18607 The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
.g              CMod

Theoremclmsubdir 18608 Scalar multiplication distributive law for subtraction. (lmodsubdir 15699 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar                     CMod

Theoremzlmclm 18609 The -module operation turns an arbitrary abelian group into a complex module. (Contributed by Mario Carneiro, 30-Oct-2015.)
Mod       CMod

Theoremclmzlmvsca 18610 The scalar product of a complex module matches the scalar product of the derived -module, which implies, together with zlmbas 16488 and zlmplusg 16489, that any module over is structure-equivalent to the canonical -module Mod. (Contributed by Mario Carneiro, 30-Oct-2015.)
Mod              CMod

Theoremnmoleub2lem 18611* Lemma for nmoleub2a 18614 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2lem3 18612* Lemma for nmoleub2a 18614 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2lem2 18613* Lemma for nmoleub2a 18614 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2a 18614* The operator norm is the supremum of the value of a linear operator in the closed unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub2b 18615* The operator norm is the supremum of the value of a linear operator in the open unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmoleub3 18616* The operator norm is the supremum of the value of a linear operator on the closed unit sphere. (Contributed by Mario Carneiro, 19-Oct-2015.)
Scalar              NrmMod CMod       NrmMod CMod       LMHom

Theoremnmhmcn 18617 A linear operator over a normed complex module is bounded iff it is continuous. (Contributed by Mario Carneiro, 22-Oct-2015.)
Scalar              NrmMod CMod NrmMod CMod NMHom LMHom

11.4.2  Complex pre-Hilbert space

Syntaxccph 18618 Extend class notation with a complex pre-Hilbert space.

Syntaxctch 18619 Function to put a norm on a Hilbert space.
toCHil

Definitiondf-cph 18620* Define a complex pre-Hilbert space. By restricting the scalar field to a quadratically closed subfield of , we have enough structure to define a norm, with the associated connection to a metric and topology. (Contributed by Mario Carneiro, 8-Oct-2015.)
NrmMod Scalar flds

Definitiondf-tch 18621* Define a function to augment a (pre-)Hilbert space with a norm. No extra parameters are needed, but some conditions must be satisfied to ensure that this in fact creates a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
toCHil toNrmGrp

Theoremiscph 18622* A complex pre-Hilbert space is a pre-Hilbert space over a quadratically closed subfield of the complexes, with a norm defined (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar              NrmMod flds

Theoremcphphl 18623 A complex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphnlm 18624 A complex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)
NrmMod

Theoremcphngp 18625 A complex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.)
NrmGrp

Theoremcphlmod 18626 A complex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphlvec 18627 A complex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphnvc 18628 A complex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.)
NrmVec

Theoremcphsubrglem 18629 Lemma for cphsubrg 18632. (Contributed by Mario Carneiro, 9-Oct-2015.)
flds               flds SubRingfld

Theoremcphreccllem 18630 Lemma for cphreccl 18633. (Contributed by Mario Carneiro, 8-Oct-2015.)
flds

Theoremcphsca 18631 A complex pre-Hilbert space is a vector space over a subfield of . (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar              flds

Theoremcphsubrg 18632 The scalar field of a complex pre-Hilbert space is a subring of . (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar              SubRingfld

Theoremcphreccl 18633 The scalar field of a complex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar

Theoremcphdivcl 18634 The scalar field of a complex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 11-Oct-2015.)
Scalar

Theoremcphcjcl 18635 The scalar field of a complex pre-Hilbert space is closed under conjugation. (Contributed by Mario Carneiro, 11-Oct-2015.)
Scalar

Theoremcphsqrcl 18636 The scalar field of a complex pre-Hilbert space is closed under square roots of positive reals (i.e. it is quadratically closed relative to ). (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar

Theoremcphabscl 18637 The scalar field of a complex pre-Hilbert space is closed under the absolute value operation. (Contributed by Mario Carneiro, 11-Oct-2015.)
Scalar

Theoremcphsqrcl2 18638 The scalar field of a complex pre-Hilbert space is closed under square roots of all numbers except possibly the negative reals. (Contributed by Mario Carneiro, 8-Oct-2015.)
Scalar

Theoremcphsqrcl3 18639 If the scalar field contains , it is completely closed under square roots (i.e. it is quadratically closed). (Contributed by Mario Carneiro, 11-Oct-2015.)
Scalar

Theoremcphqss 18640 The scalar field of a complex pre-Hilbert space contains all rational numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
Scalar

Theoremcphclm 18641 A complex pre-Hilbert space is a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
CMod

Theoremcphnmvs 18642 Norm of a scalar product. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar

Theoremcphipcl 18643 An inner product is a member of the complex numbers. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremcphnmfval 18644* The value of the norm in a complex pre-Hilbert space is the square root of the inner product of a vector with itself. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphnm 18645 The square of the norm is the norm of an inner product in a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremnmsq 18646 The square of the norm is the norm of an inner product in a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphnmf 18647 The norm of a vector is a member of the scalar field in a complex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
Scalar

Theoremcphnmcl 18648 The norm of a vector is a member of the scalar field in a complex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
Scalar

Theoremreipcl 18649 An inner product of an element with itself is real. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremipge0 18650 The inner product in a complex pre-Hilbert space is positive definite. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremcphipcj 18651 Conjugate of an inner product in a complex pre-Hilbert space. Complex version of ipcj 16554. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theoremcphorthcom 18652 Orthogonality (meaning inner product is 0) is commutative. Complex version of iporthcom 16555. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theoremcphip0l 18653 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. Complex version of ip0l 16556. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theoremcphip0r 18654 Inner product with a zero second argument. Complex version of ip0r 16557. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theoremcphipeq0 18655 The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. Complex version of ipeq0 16558. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theoremcphdir 18656 Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362. Complex version of ipdir 16559. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theoremcphdi 18657 Distributive law for inner product. Complex version of ipdi 16560. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theoremcph2di 18658 Distributive law for inner product. Complex version of ip2di 16561. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theoremcphsubdir 18659 Distributive law for inner product subtraction. Complex version of ipsubdir 16562. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theoremcphsubdi 18660 Distributive law for inner product subtraction. Complex version of ipsubdi 16563. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theoremcph2subdi 18661 Distributive law for inner product subtraction. Complex version of ip2subdi 16564. (Contributed by Mario Carneiro, 16-Oct-2015.)

Theoremcphass 18662 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. See ipass 16565, his5 21681. (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar

Theoremcphassr 18663 "Associative" law for second argument of inner product (compare cphass 18662). See ipassr 16566, his52 . (Contributed by Mario Carneiro, 16-Oct-2015.)
Scalar

Theoremcph2ass 18664 Move scalar multiplication to outside of inner product. See his35 21683. (Contributed by Mario Carneiro, 17-Oct-2015.)
Scalar

Theoremtchex 18665* Lemma for tchbas 18667 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremtchval 18666* Define a function to augment a pre-Hilbert space with norm. (Contributed by Mario Carneiro, 7-Oct-2015.)
toCHil                     toNrmGrp

Theoremtchbas 18667 The base set of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
toCHil

Theoremtchplusg 18668 The addition operation of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
toCHil

Theoremtchmulr 18669 The ring operation of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
toCHil

Theoremtchsca 18670 The scalar field of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
toCHil       Scalar       Scalar

Theoremtchvsca 18671 The scalar multiplication of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
toCHil

Theoremtchip 18672 The inner product of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
toCHil

Theoremtchtopn 18673 The topology of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
toCHil

Theoremtchphl 18674 Augmentation of a pre-Hilbert space with a norm does not affect whether it is still a pre-Hilbert space because all the orginal components are the same. (Contributed by Mario Carneiro, 8-Oct-2015.)
toCHil

Theoremtchnmfval 18675* The norm of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
toCHil

Theoremtchnmval 18676 The norm of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
toCHil

Theoremcphtchnm 18677 The norm of a norm-augmented complex pre-Hilbert space is the same as the original norm on it. (Contributed by Mario Carneiro, 11-Oct-2015.)
toCHil

Theoremtchclm 18678 Lemma for tchcph 18683. (Contributed by Mario Carneiro, 16-Oct-2015.)
toCHil              Scalar              flds        CMod

Theoremtchcphlem3 18679 Lemma for tchcph 18683: real closure of an inner product of a vector with itself. (Contributed by Mario Carneiro, 10-Oct-2015.)
toCHil              Scalar              flds

Theoremipcau2 18680* The Cauchy-Schwarz inequality for a complex pre-Hilbert space. (Contributed by Mario Carneiro, 11-Oct-2015.)
toCHil              Scalar              flds

Theoremtchcphlem1 18681* Lemma for tchcph 18683: the triangle inequality. (Contributed by Mario Carneiro, 8-Oct-2015.)
toCHil              Scalar              flds

Theoremtchcphlem2 18682* Lemma for tchcph 18683: homogeneity. (Contributed by Mario Carneiro, 8-Oct-2015.)
toCHil              Scalar              flds

Theoremtchcph 18683* The standard definition of a norm turns any pre-Hilbert space over a quadratically closed subfield of into a complex pre-Hilbert space (which allows access to a norm, metric, and topology). (Contributed by Mario Carneiro, 11-Oct-2015.)
toCHil              Scalar              flds

Theoremipcau 18684 The Cauchy-Schwarz inequality for a complex pre-Hilbert space. (Contributed by Mario Carneiro, 11-Oct-2015.)

Theoremnmparlem 18685 Lemma for nmpar 18686. (Contributed by Mario Carneiro, 7-Oct-2015.)
Scalar

Theoremnmpar 18686 A complex pre-Hilbert space satisfies the parallelogram law. (Contributed by Mario Carneiro, 7-Oct-2015.)

Theoremipcnlem2 18687 The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremipcnlem1 18688* The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremipcn 18689 The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
fld

Theoremcnmpt1ip 18690* Continuity of inner product; analogue of cnmpt12f 17376 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)
fld                     TopOn

Theoremcnmpt2ip 18691* Continuity of inner product; analogue of cnmpt22f 17385 which cannot be used directly because is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)
fld                     TopOn       TopOn

Theoremcsscld 18692 A "closed subspace" in a complex pre-Hilbert space is actually closed in the topology induced by the norm, thus justifying the terminology "closed subspace". (Contributed by Mario Carneiro, 13-Oct-2015.)

Theoremclsocv 18693 The orthogonal complement of the closure of a subset is the same as the orthogonal complement of the subset itself. (Contributed by Mario Carneiro, 13-Oct-2015.)

11.4.3  Convergence and completeness

Syntaxccfil 18694 Extend class notation with the set of Cauchy filters.
CauFil

Syntaxcca 18695 Extend class notation with a function on metric spaces whose value is the set of all Cauchy sequences of the space.

Syntaxcms 18696 Extend class notation with class of complete metric spaces.

Definitiondf-cfil 18697* Define the set of Cauchy filters on a metric space. A Cauchy filter is a filter on the set such that for every there is an element of the filter whose metric diameter is less than . (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil

Definitiondf-cau 18698* Define a function on metric spaces whose value is the set of Cauchy sequences of the space. (Contributed by NM, 8-Sep-2006.)

Definitiondf-cmet 18699* Define the class of complete metrics. (Contributed by Mario Carneiro, 1-May-2014.)
CauFil

Theoremlmmbr 18700* Express the binary relation "sequence converges to point " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The condition allows us to use objects more general than sequences when convenient; see the comment in df-lm 16975. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)

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 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32776
 Copyright terms: Public domain < Previous  Next >