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Theorem List for Metamath Proof Explorer - 23401-23500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremspecval 23401* 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 23402 The spectrum of an operator is a set of complex numbers. (Contributed by NM, 11-Apr-2006.) (New usage is discouraged.)

Theoremhhlnoi 23403 The linear operators of Hilbert space. (Contributed by NM, 19-Nov-2007.) (Revised by Mario Carneiro, 19-Nov-2013.) (New usage is discouraged.)

Theoremhhnmoi 23404 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 23405 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 23406 The zero operator in Hilbert space. (Contributed by NM, 7-Dec-2007.) (New usage is discouraged.)

Theoremhhcno 23407 The continuous operators of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)

Theoremhhcnf 23408 The continuous functionals of Hilbert space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
fld

Theoremdmadjrnb 23409 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 5755.) (Contributed by NM, 19-Feb-2006.) (New usage is discouraged.)

Theoremnmoplb 23410 A lower bound for an operator norm. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Theoremnmopub 23411* An upper bound for an operator norm. (Contributed by NM, 7-Mar-2006.) (New usage is discouraged.)

Theoremnmopub2tALT 23412* An upper bound for an operator norm. (Contributed by NM, 12-Apr-2006.) (New usage is discouraged.)

Theoremnmopub2tHIL 23413* An upper bound for an operator norm. (Contributed by NM, 13-Dec-2007.) (New usage is discouraged.)

Theoremnmopge0 23414 The norm of any Hilbert space operator is nonnegative. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)

Theoremnmopgt0 23415 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 23416* 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 23417 Basic linearity property of a linear Hilbert space operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

Theoremunop 23418 Basic inner product property of a unitary operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

Theoremunopf1o 23419 A unitary operator in Hilbert space is one-to-one and onto. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

Theoremunopnorm 23420 A unitary operator is idempotent in the norm. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.)

Theoremcnvunop 23421 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 23422 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 23423 A unitary operator is linear. Theorem in [AkhiezerGlazman] p. 72. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

Theoremcounop 23424 The composition of two unitary operators is unitary. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.)

Theoremhmop 23425 Basic inner product property of a Hermitian operator. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)

Theoremhmopre 23426 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 23427 A lower bound for a functional norm. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Theoremnmfnleub 23428* 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 23429* An upper bound for the norm of a functional. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)

Theoremnmfnge0 23430 The norm of any Hilbert space functional is nonnegative. (Contributed by NM, 24-May-2006.) (New usage is discouraged.)

Theoremelnlfn 23431 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 23432 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 23433* 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 23434 Basic linearity property of a linear functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.)

Theoremadjcl 23435 Closure of the adjoint of a Hilbert space operator. (Contributed by NM, 17-Jun-2006.) (New usage is discouraged.)

Theoremadj1 23436 Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremadj2 23437 Property of an adjoint Hilbert space operator. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremadjeq 23438* A property that determines the adjoint of a Hilbert space operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremadjadj 23439 Double adjoint. Theorem 3.11(iv) of [Beran] p. 106. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

Theoremadjvalval 23440* 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 23441 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 23442 A Hermitian operator is self-adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theoremhmdmadj 23443 Every Hermitian operator has an adjoint. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theoremhmopadj2 23444 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 23445 A Hermitian operator is linear. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Theorembrafval 23446* The bra of a vector, expressed as in Dirac notation. See df-bra 23353. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Theorembraval 23447 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.)

Theorembraadd 23448 Linearity property of bra for addition. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Theorembramul 23449 Linearity property of bra for multiplication. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Theorembrafn 23450 The bra function is a functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theorembralnfn 23451 The Dirac bra function is a linear functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theorembracl 23452 Closure of the bra function. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

Theorembra0 23453 The Dirac bra of the zero vector. (Contributed by NM, 25-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Theorembrafnmul 23454 Anti-linearity property of bra functional for multiplication. (Contributed by NM, 31-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremkbfval 23455* The outer product of two vectors, expressed as in Dirac notation. See df-kb 23354. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

Theoremkbop 23456 The outer product of two vectors, expressed as in Dirac notation, is an operator. (Contributed by NM, 30-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremkbval 23457 The value of the operator resulting from the outer product of two vectors. Equation 8.1 of [Prugovecki] p. 376. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Theoremkbmul 23458 Multiplication property of outer product. (Contributed by NM, 31-May-2006.) (New usage is discouraged.)

Theoremkbpj 23459 If a vector has norm 1, the outer product is the projector onto the subspace spanned by . http://en.wikipedia.org/wiki/Bra-ket#Linear%5Foperators. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)

Theoremeleigvec 23460* Membership in 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.)

Theoremeleigvec2 23461 Membership in the set of eigenvectors of a Hilbert space operator. (Contributed by NM, 18-Mar-2006.) (New usage is discouraged.)

Theoremeleigveccl 23462 Closure of an eigenvector of a Hilbert space operator. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)

Theoremeigvalval 23463 The eigenvalue of an eigenvector of a Hilbert space operator. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)

Theoremeigvalcl 23464 An eigenvalue is a complex number. (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)

Theoremeigvec1 23465 Property of an eigenvector. (Contributed by NM, 12-Mar-2006.) (New usage is discouraged.)

Theoremeighmre 23466 The eigenvalues of a Hermitian operator are real. Equation 1.30 of [Hughes] p. 49. (Contributed by NM, 19-Mar-2006.) (New usage is discouraged.)

Theoremeighmorth 23467 Eigenvectors of a Hermitian operator with distinct eigenvalues are orthogonal. Equation 1.31 of [Hughes] p. 49. (Contributed by NM, 23-Mar-2006.) (New usage is discouraged.)

Theoremnmopnegi 23468 Value of the norm of the negative of a Hilbert space operator. Unlike nmophmi 23534, the operator does not have to be bounded. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.)

Theoremlnop0 23469 The value of a linear Hilbert space operator at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)

Theoremlnopmul 23470 Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)

Theoremlnopli 23471 Basic scalar product property of a linear Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.)

Theoremlnopfi 23472 A linear Hilbert space operator is a Hilbert space operator. (Contributed by NM, 23-Jan-2006.) (New usage is discouraged.)

Theoremlnop0i 23473 The value of a linear Hilbert space operator at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)

Theoremlnopaddi 23474 Additive property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)

Theoremlnopmuli 23475 Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.)

Theoremlnopaddmuli 23476 Sum/product property of a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)

Theoremlnopsubi 23477 Subtraction property for a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)

Theoremlnopsubmuli 23478 Subtraction/product property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)

Theoremlnopmulsubi 23479 Product/subtraction property of a linear Hilbert space operator. (Contributed by NM, 2-Jul-2005.) (New usage is discouraged.)

Theoremhomco2 23480 Move a scalar product out of a composition of operators. The operator must be linear, unlike homco1 23304 that works for any operators. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.)

Theoremidunop 23481 The identity function (restricted to Hilbert space) is a unitary operator. (Contributed by NM, 21-Jan-2006.) (New usage is discouraged.)

Theorem0cnop 23482 The identically zero function is a continuous Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Theorem0cnfn 23483 The identically zero function is a continuous Hilbert space functional. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Theoremidcnop 23484 The identity function (restricted to Hilbert space) is a continuous operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Theoremidhmop 23485 The Hilbert space identity operator is a Hermitian operator. (Contributed by NM, 22-Apr-2006.) (New usage is discouraged.)

Theorem0hmop 23486 The identically zero function is a Hermitian operator. (Contributed by NM, 8-Aug-2006.) (New usage is discouraged.)

Theorem0lnop 23487 The identically zero function is a linear Hilbert space operator. (Contributed by NM, 7-Feb-2006.) (New usage is discouraged.)

Theorem0lnfn 23488 The identically zero function is a linear Hilbert space functional. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Theoremnmop0 23489 The norm of the zero operator is zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.)

Theoremnmfn0 23490 The norm of the identically zero functional is zero. (Contributed by NM, 25-Apr-2006.) (New usage is discouraged.)

TheoremhmopbdoptHIL 23491 A Hermitian operator is a bounded linear operator (Hellinger-Toeplitz Theorem). (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.)

Theoremhoddii 23492 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 23283 does not require linearity.) (Contributed by NM, 11-Mar-2006.) (New usage is discouraged.)

Theoremhoddi 23493 Distributive law for Hilbert space operator difference. (Interestingly, the reverse distributive law hocsubdiri 23283 does not require linearity.) (Contributed by NM, 23-Aug-2006.) (New usage is discouraged.)

Theoremnmop0h 23494 The norm of any operator on the trivial Hilbert space is zero. (This is the reason we need in nmopun 23517.) (Contributed by NM, 24-Feb-2006.) (New usage is discouraged.)

Theoremidlnop 23495 The identity function (restricted to Hilbert space) is a linear operator. (Contributed by NM, 24-Jan-2006.) (New usage is discouraged.)

Theorem0bdop 23496 The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.)

Theoremadj0 23497 Adjoint of the zero operator. (Contributed by NM, 20-Feb-2006.) (New usage is discouraged.)

Theoremnmlnop0iALT 23498 A linear operator with a zero norm is identically zero. (Contributed by NM, 8-Feb-2006.) (New usage is discouraged.)

Theoremnmlnop0iHIL 23499 A linear operator with a zero norm is identically zero. (Contributed by NM, 18-Jan-2008.) (New usage is discouraged.)

Theoremnmlnopgt0i 23500 A linear Hilbert space operator that is not identically zero has a positive norm. (Contributed by NM, 9-Feb-2006.) (New usage is discouraged.)

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