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Theorem List for Metamath Proof Explorer - 22201-22300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhoaddid1 22201 Sum of a Hilbert space operator with the zero operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)

Theoremhodid 22202 Difference of a Hilbert space operator from itself. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Theoremhon0 22203 A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theoremhodseqi 22204 Subtraction and addition of equal Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)

Theoremho0subi 22205 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theoremhonegsubi 22206 Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremho0sub 22207 Subtraction of Hilbert space operators expressed in terms of difference from zero. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)

Theoremhosubid1 22208 The zero operator subtracted from a Hilbert space operator. (Contributed by NM, 25-Jul-2006.) (New usage is discouraged.)

Theoremhonegsub 22209 Relationship between Hilbert space operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhomulid2 22210 An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremhomco1 22211 Associative law for scalar product and composition of operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)

Theoremhomulass 22212 Scalar product associative law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremhoadddi 22213 Scalar product distributive law for Hilbert space operators. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremhoadddir 22214 Scalar product reverse distributive law for Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)

Theoremhomul12 22215 Swap first and second factors in a nested operator scalar product. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremhonegneg 22216 Double negative of a Hilbert space operator. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhosubneg 22217 Relationship between operator subtraction and negative. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)

Theoremhosubdi 22218 Scalar product distributive law for operator difference. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)

Theoremhonegdi 22219 Distribution of negative over addition. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhonegsubdi 22220 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhonegsubdi2 22221 Distribution of negative over subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhosubsub2 22222 Law for double subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhosub4 22223 Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhosubadd4 22224 Rearrangement of 4 terms in a mixed addition and subtraction of Hilbert space operators. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.)

Theoremhoaddsubass 22225 Associative-type law for addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)

Theoremhoaddsub 22226 Law for operator addition and subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)

Theoremhosubsub 22227 Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)

Theoremhosubsub4 22228 Law for double subtraction of Hilbert space operators. (Contributed by NM, 25-Aug-2006.) (New usage is discouraged.)

Theoremho2times 22229 Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006.) (New usage is discouraged.)

Theoremhoaddsubassi 22230 Associativity of sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)

Theoremhoaddsubi 22231 Law for sum and difference of Hilbert space operators. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)

Theoremhosd1i 22232 Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)

Theoremhosd2i 22233 Hilbert space operator sum expressed in terms of difference. (Contributed by NM, 27-Aug-2004.) (New usage is discouraged.)

Theoremhopncani 22234 Hilbert space operator cancellation law. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theoremhonpcani 22235 Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)

Theoremhosubeq0i 22236 If the difference between two operators is zero, they are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.)

Theoremhonpncani 22237 Hilbert space operator cancellation law. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)

Theoremho01i 22238* A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S8) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)

Theoremho02i 22239* A condition implying that a Hilbert space operator is identically zero. Lemma 3.2(S10) of [Beran] p. 95. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)

Theoremhoeq1 22240* A condition implying that two Hilbert space operators are equal. Lemma 3.2(S9) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremhoeq2 22241* A condition implying that two Hilbert space operators are equal. Lemma 3.2(S11) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremadjmo 22242* Every Hilbert space operator has at most one adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)

Theoremadjsym 22243* Symmetry property of an adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)

Theoremeigrei 22244 A necessary and sufficient condition (that holds when is a Hermitian operator) for an eigenvalue to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 21-Jan-2005.) (New usage is discouraged.)

Theoremeigre 22245 A necessary and sufficient condition (that holds when is a Hermitian operator) for an eigenvalue to be real. Generalization of Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)

Theoremeigposi 22246 A sufficient condition (first conjunct pair, that holds when is a positive operator) for an eigenvalue (second conjunct pair) to be nonnegative. Remark (ii) in [Hughes] p. 137. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)

Theoremeigorthi 22247 A necessary and sufficient condition (that holds when is a Hermitian operator) for two eigenvectors and to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Jan-2005.) (New usage is discouraged.)

Theoremeigorth 22248 A necessary and sufficient condition (that holds when is a Hermitian operator) for two eigenvectors and to be orthogonal. Generalization of Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)

15.9.35  Linear, continuous, bounded, Hermitian, unitary operators and norms

Definitiondf-nmop 22249* Define the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)

Definitiondf-cnop 22250* Define the set of continuous operators on Hilbert space. For every "epsilon" () there is an "delta" () such that... (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)

Definitiondf-lnop 22251* Define the set of linear operators on Hilbert space. (See df-hosum 21992 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)

Definitiondf-bdop 22252 Define the set of bounded linear Hilbert space operators. (See df-hosum 21992 for definition of operator.) (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)

Definitiondf-unop 22253* Define the set of unitary operators on Hilbert space. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)

Definitiondf-hmop 22254* Define the set of Hermitian operators on Hilbert space. Some books call these "symmetric operators" and others call them "self-adjoint operators," sometimes with slightly different technical meanings. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)

15.9.36  Linear and continuous functionals and norms

Definitiondf-nmfn 22255* Define the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Definitiondf-nlfn 22256 Define the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Definitiondf-cnfn 22257* Define the set of continuous functionals on Hilbert space. For every "epsilon" () there is an "delta" () such that... (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Definitiondf-lnfn 22258* Define the set of linear functionals on Hilbert space. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Definitiondf-adjh 22259* Define the adjoint of a Hilbert space operator (if it exists). The domain of is the set of all adjoint operators. Definition of adjoint in [Kalmbach2] p. 8. Unlike Kalmbach (and most authors), we do not demand that the operator be linear, but instead show (in adjbdln 22493) that the adjoint exists for a bounded linear operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

15.9.38  Dirac bra-ket notation

Definitiondf-bra 22260* Define the bra of a vector used by Dirac notation. Based on definition of bra in [Prugovecki] p. 186 (p. 180 in 1971 edition). In Dirac bra-ket notation, is a complex number equal to the inner product . But physicists like to talk about the individual components and , called bra and ket respectively. In order for their properties to make sense formally, we define the ket as the vector itself, and the bra as a functional from to . We represent the Dirac notation by ; see braval 22354. The reversal of the inner product arguments not only makes the bra-ket behavior consistent with physics literature (see comments under ax-his3 21493) but is also required in order for the associative law kbass2 22527 to work.

Our definition of bra and the associated outer product df-kb 22261 differs from, but is equivalent to, a common approach in the literature that makes use of mappings to a dual space. Our approach eliminates the need to have a parallel development of this dual space and instead keeps everything in Hilbert space.

For an extensive discussion about how our notation maps to the bra-ket notation in physics textbooks, see http://us.metamath.org/mpegif/mmnotes.txt, under the 17-May-2006 entry. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)

Definitiondf-kb 22261* Define a commuted bra and ket juxtaposition used by Dirac notation. In Dirac notation, is an operator known as the outer product of and , which we represent by . Based on Equation 8.1 of [Prugovecki] p. 376. This definition, combined with definition df-bra 22260, allows any legal juxtaposition of bras and kets to make sense formally and also to obey the associative law when mapped back to Dirac notation. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)

15.9.39  Positive operators

Definitiondf-leop 22262* Define positive operator ordering. Definition VI.1 of [Retherford] p. 49. Note that means that is a positive operator. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

15.9.40  Eigenvectors, eigenvalues, spectrum

Definitiondf-eigvec 22263* Define the eigenvector function. Theorem eleigveccl 22369 shows that , the set of eigenvectors of Hilbert space operator , are Hilbert space vectors. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)

Definitiondf-eigval 22264* Define the eigenvalue function. The range of is the set of eigenvalues of Hilbert space operator . Theorem eigvalcl 22371 shows that , the eigenvalue associated with eigenvector , is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)

Definitiondf-spec 22265* Define the spectrum of an operator. Definition of spectrum in [Halmos] p. 50. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)

15.9.41  Theorems about operators and functionals

Theoremnmopval 22266* Value of the norm of a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremelcnop 22267* Property defining a continuous Hilbert space operator. (Contributed by NM, 28-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremellnop 22268* Property defining a linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremlnopf 22269 A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)

Theoremelbdop 22270 Property defining a bounded linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theorembdopln 22271 A bounded linear Hilbert space operator is a linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)

Theorembdopf 22272 A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)

TheoremnmopsetretALT 22273* The set in the supremum of the operator norm definition df-nmop 22249 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)

TheoremnmopsetretHIL 22274* The set in the supremum of the operator norm definition df-nmop 22249 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)

Theoremnmopsetn0 22275* The set in the supremum of the operator norm definition df-nmop 22249 is nonempty. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)

Theoremnmopxr 22276 The norm of a Hilbert space operator is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)

Theoremnmoprepnf 22277 The norm of a Hilbert space operator is either real or plus infinity. (Contributed by NM, 5-Feb-2006.) (New usage is discouraged.)

Theoremnmopgtmnf 22278 The norm of a Hilbert space operator is not minus infinity. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)

Theoremnmopreltpnf 22279 The norm of a Hilbert space operator is real iff it is less than infinity. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Theoremnmopre 22280 The norm of a bounded operator is a real number. (Contributed by NM, 29-Jan-2006.) (New usage is discouraged.)

Theoremelbdop2 22281 Property defining a bounded linear Hilbert space operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Theoremelunop 22282* Property defining a unitary Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)

Theoremelhmop 22283* Property defining a Hermitian Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremhmopf 22284 A Hermitian operator is a Hilbert space operator (mapping). (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)

Theoremhmopex 22285 The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)

Theoremnmfnval 22286* Value of the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremnmfnsetre 22287* The set in the supremum of the functional norm definition df-nmfn 22255 is a set of reals. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Theoremnmfnsetn0 22288* The set in the supremum of the functional norm definition df-nmfn 22255 is nonempty. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Theoremnmfnxr 22289 The norm of any Hilbert space functional is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)

Theoremnmfnrepnf 22290 The norm of a Hilbert space functional is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theoremnlfnval 22291 Value of the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Theoremelcnfn 22292* Property defining a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremellnfn 22293* Property defining a linear functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremlnfnf 22294 A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)

Theoremdfadj2 22295* Alternate definition of the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremfunadj 22296 Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremdmadjss 22297 The domain of the adjoint function is a subset of the maps from to . (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremdmadjop 22298 A member of the domain of the adjoint function is a Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremadjeu 22299* Elementhood in the domain of the adjoint function. (Contributed by Mario Carneiro, 11-Sep-2015.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.)

Theoremadjval 22300* Value of the adjoint function for in the domain of . (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.)

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