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18.6.5  Linear and continuous functionals and norms

Definitiondf-nmfn 23301* Define the norm of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Definitiondf-nlfn 23302 Define the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Definitiondf-cnfn 23303* Define the set of continuous functionals on Hilbert space. For every "epsilon" () there is a "delta" () such that... (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Definitiondf-lnfn 23304* Define the set of linear functionals on Hilbert space. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Definitiondf-adjh 23305* 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 23539) that the adjoint exists for a bounded linear operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

18.6.7  Dirac bra-ket notation

Definitiondf-bra 23306* 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 23400. The reversal of the inner product arguments not only makes the bra-ket behavior consistent with physics literature (see comments under ax-his3 22539) but is also required in order for the associative law kbass2 23573 to work.

Our definition of bra and the associated outer product df-kb 23307 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/mpeuni/mmnotes.txt, under the 17-May-2006 entry. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)

Definitiondf-kb 23307* 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 23306, 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.)

18.6.8  Positive operators

Definitiondf-leop 23308* 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.)

18.6.9  Eigenvectors, eigenvalues, spectrum

Definitiondf-eigvec 23309* Define the eigenvector function. Theorem eleigveccl 23415 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 23310* Define the eigenvalue function. The range of is the set of eigenvalues of Hilbert space operator . Theorem eigvalcl 23417 shows that , the eigenvalue associated with eigenvector , is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)

Definitiondf-spec 23311* Define the spectrum of an operator. Definition of spectrum in [Halmos] p. 50. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)

18.6.10  Theorems about operators and functionals

Theoremnmopval 23312* 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 23313* 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 23314* 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 23315 A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)

Theoremelbdop 23316 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 23317 A bounded linear Hilbert space operator is a linear operator. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)

Theorembdopf 23318 A bounded linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)

TheoremnmopsetretALT 23319* The set in the supremum of the operator norm definition df-nmop 23295 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)

TheoremnmopsetretHIL 23320* The set in the supremum of the operator norm definition df-nmop 23295 is a set of reals. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)

Theoremnmopsetn0 23321* The set in the supremum of the operator norm definition df-nmop 23295 is nonempty. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)

Theoremnmopxr 23322 The norm of a Hilbert space operator is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)

Theoremnmoprepnf 23323 The norm of a Hilbert space operator is either real or plus infinity. (Contributed by NM, 5-Feb-2006.) (New usage is discouraged.)

Theoremnmopgtmnf 23324 The norm of a Hilbert space operator is not minus infinity. (Contributed by NM, 2-Feb-2006.) (New usage is discouraged.)

Theoremnmopreltpnf 23325 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 23326 The norm of a bounded operator is a real number. (Contributed by NM, 29-Jan-2006.) (New usage is discouraged.)

Theoremelbdop2 23327 Property defining a bounded linear Hilbert space operator. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Theoremelunop 23328* Property defining a unitary Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (New usage is discouraged.)

Theoremelhmop 23329* 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 23330 A Hermitian operator is a Hilbert space operator (mapping). (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)

Theoremhmopex 23331 The class of Hermitian operators is a set. (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.)

Theoremnmfnval 23332* 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 23333* The set in the supremum of the functional norm definition df-nmfn 23301 is a set of reals. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Theoremnmfnsetn0 23334* The set in the supremum of the functional norm definition df-nmfn 23301 is nonempty. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Theoremnmfnxr 23335 The norm of any Hilbert space functional is an extended real. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)

Theoremnmfnrepnf 23336 The norm of a Hilbert space functional is either real or plus infinity. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.)

Theoremnlfnval 23337 Value of the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Theoremelcnfn 23338* Property defining a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremellnfn 23339* Property defining a linear functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremlnfnf 23340 A linear Hilbert space functional is a functional. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)

Theoremdfadj2 23341* Alternate definition of the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremfunadj 23342 Functionality of the adjoint function. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremdmadjss 23343 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 23344 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 23345* 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 23346* 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.)

Theoremadjval2 23347* Value of the adjoint function. (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)

Theoremcnvadj 23348 The adjoint function equals its converse. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremfuncnvadj 23349 The converse of the adjoint function is a function. (Contributed by NM, 25-Jan-2006.) (New usage is discouraged.)

Theoremadj1o 23350 The adjoint function maps one-to-one onto its domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremdmadjrn 23351 The adjoint of an operator belongs to the adjoint function's domain. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremeigvecval 23352* The set of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremeigvalfval 23353* The eigenvalues of eigenvectors of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)

Theoremspecval 23354* The value of the spectrum of an operator. (Contributed by NM, 11-Apr-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremspeccl 23355 The spectrum of an operator is a set of complex numbers. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)

Theoremhhlnoi 23356 The linear operators of Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremhhnmoi 23357 The norm of an operator in Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)

Theoremhhbloi 23358 A bounded linear operator in Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremhh0oi 23359 The zero operator in Hilbert space. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)

Theoremhhcno 23360 The continuous operators of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theoremhhcnf 23361 The continuous functionals of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
fld

Theoremdmadjrnb 23362 The adjoint of an operator belongs to the adjoint function's domain. (Note: the converse is dependent on our definition of function value, since it uses ndmfv 5714.) (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)

Theoremnmoplb 23363 A lower bound for an operator norm. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Theoremnmopub 23364* An upper bound for an operator norm. (Contributed by NM, 7-Mar-2006.) (New usage is discouraged.)

Theoremnmopub2tALT 23365* An upper bound for an operator norm. (Contributed by NM, 12-Apr-2006.) (New usage is discouraged.)

Theoremnmopub2tHIL 23366* An upper bound for an operator norm. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)

Theoremnmopge0 23367 The norm of any Hilbert space operator is nonnegative. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)

Theoremnmopgt0 23368 A linear Hilbert space operator that is not identically zero has a positive norm. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)

Theoremcnopc 23369* Basic continuity property of a continuous Hilbert space operator. (Contributed by NM, 2-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremlnopl 23370 Basic linearity property of a linear Hilbert space operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

Theoremunop 23371 Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

Theoremunopf1o 23372 A unitary operator in Hilbert space is one-to-one and onto. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

Theoremunopnorm 23373 A unitary operator is idempotent in the norm. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Theoremcnvunop 23374 The inverse (converse) of a unitary operator in Hilbert space is unitary. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

Theoremunopadj 23375 The inverse (converse) of a unitary operator is its adjoint. Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

Theoremunoplin 23376 A unitary operator is linear. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

Theoremcounop 23377 The composition of two unitary operators is unitary. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

Theoremhmop 23378 Basic inner product property of a Hermitian operator. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)

Theoremhmopre 23379 The inner product of the value and argument of a Hermitian operator is real. (Contributed by NM, 23-Jul-2006.) (New usage is discouraged.)

Theoremnmfnlb 23380 A lower bound for a functional norm. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Theoremnmfnleub 23381* An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (Revised by Mario Carneiro, 7-Sep-2014.) (New usage is discouraged.)

Theoremnmfnleub2 23382* An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)

Theoremnmfnge0 23383 The norm of any Hilbert space functional is nonnegative. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)

Theoremelnlfn 23384 Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)

Theoremelnlfn2 23385 Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)

Theoremcnfnc 23386* Basic continuity property of a continuous functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremlnfnl 23387 Basic linearity property of a linear functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Theoremadjcl 23388 Closure of the adjoint of a Hilbert space operator. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)

Theoremadj1 23389 Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremadj2 23390 Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremadjeq 23391* A property that determines the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremadjadj 23392 Double adjoint. Theorem 3.11(iv) of [Beran] p. 106. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremadjvalval 23393* Value of the value of the adjoint function. (Contributed by NM, 22-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

Theoremunopadj2 23394 The adjoint of a unitary operator is its inverse (converse). Equation 2 of [AkhiezerGlazman] p. 72. (Contributed by NM, 23-Feb-2006.) (New usage is discouraged.)

Theoremhmopadj 23395 A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theoremhmdmadj 23396 Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theoremhmopadj2 23397 An operator is Hermitian iff it is self-adjoint. Definition of Hermitian in [Halmos] p. 41. (Contributed by NM, 9-Apr-2006.) (New usage is discouraged.)

Theoremhmoplin 23398 A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theorembrafval 23399* The bra of a vector, expressed as in Dirac notation. See df-bra 23306. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Theorembraval 23400 A bra-ket juxtaposition, expressed as in Dirac notation, equals the inner product of the vectors. Based on definition of bra in [Prugovecki] p. 186. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)

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