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Theorem List for Metamath Proof Explorer - 22401-22500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhlipcj 22401 Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipdir 22402 Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipass 22403 Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipgt0 22404 The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlcompl 22405 Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)

17.6.3  Examples of complex Hilbert spaces

Theoremcnchl 22406 The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

17.6.4  Subspaces

Theoremssphl 22407 A Banach subspace of an inner product space is a Hilbert space. (Contributed by NM, 11-Apr-2008.) (New usage is discouraged.)

17.6.5  Hellinger-Toeplitz Theorem

Theoremhtthlem 22408* Lemma for htth 22409. The collection , which consists of functions for each in the unit ball, is a collection of bounded linear functions by ipblnfi 22345, so by the Uniform Boundedness theorem ubth 22363, there is a uniform bound on for all in the unit ball. Then , so and is bounded. (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
CV

Theoremhtth 22409* Hellinger-Toeplitz Theorem: any self-adjoint linear operator defined on all of Hilbert space is bounded. Theorem 10.1-1 of [Kreyszig] p. 525. Discovered by E. Hellinger and O. Toeplitz in 1910, "it aroused both admiration and puzzlement since the theorem establishes a relation between properties of two different kinds, namely, the properties of being defined everywhere and being bounded." (Contributed by NM, 11-Jan-2008.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)

PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)

This part contains the definitions and theorems used by the Hilbert Space Explorer http://us.metamath.org/mpeuni/mmhil.html. Because it axiomatizes a single complex Hilbert space whose existence is assumed, its usefulness is limited. For example, it cannot work with real or quaternion Hilbert spaces and it cannot study relationships between two Hilbert spaces. More information can be found on the Hilbert Space Explorer page. Future development should work with general Hilbert spaces as defined by df-hil 16919.

18.1  Axiomatization of complex pre-Hilbert spaces

18.1.1  Basic Hilbert space definitions

Syntaxchil 22410 Extend class notation with Hilbert vector space.

Syntaxcva 22411 Extend class notation with vector addition in Hilbert space. In the literature, the subscript "v" is omitted, but we need it to avoid ambiguity with complex number addition caddc 8982.

Syntaxcsm 22412 Extend class notation with scalar multiplication in Hilbert space. In the literature scalar multiplication is usually indicated by juxtaposition, but we need an explicit symbol to prevent ambiguity.

Syntaxcsp 22413 Extend class notation with inner (scalar) product in Hilbert space. In the literature, the inner product of and is usually written but our operation notation allows us to use existing theorems about operations and also eliminates ambiguity with the definition of an ordered pair df-op 3815.

Syntaxcno 22414 Extend class notation with the norm function in Hilbert space. In the literature, the norm of is usually written "|| ||", but we use function notation to take advantage of our existing theorems about functions.

Syntaxc0v 22415 Extend class notation with zero vector in Hilbert space.

Syntaxcmv 22416 Extend class notation with vector subtraction in Hilbert space.

Syntaxccau 22417 Extend class notation with set of Cauchy sequences in Hilbert space.

Syntaxchli 22418 Extend class notation with convergence relation in Hilbert space.

Syntaxcsh 22419 Extend class notation with set of subspaces of a Hilbert space.

Syntaxcch 22420 Extend class notation with set of closed subspaces of a Hilbert space.

Syntaxcort 22421 Extend class notation with orthogonal complement in .

Syntaxcph 22422 Extend class notation with subspace sum in .

Syntaxcspn 22423 Extend class notation with subspace span in .

Syntaxchj 22424 Extend class notation with join in .

Syntaxchsup 22425 Extend class notation with supremum of a collection in .

Syntaxc0h 22426 Extend class notation with zero of .

Syntaxccm 22427 Extend class notation with the commutes relation on a Hilbert lattice.

Syntaxcpjh 22428 Extend class notation with set of projections on a Hilbert space.

Syntaxchos 22429 Extend class notation with sum of Hilbert space operators.

Syntaxchot 22430 Extend class notation with scalar product of a Hilbert space operator.

Syntaxchod 22431 Extend class notation with difference of Hilbert space operators.

Syntaxchfs 22432 Extend class notation with sum of Hilbert space functionals.

Syntaxchft 22433 Extend class notation with scalar product of Hilbert space functional.

Syntaxch0o 22434 Extend class notation with the Hilbert space zero operator.

Syntaxchio 22435 Extend class notation with Hilbert space identity operator.

Syntaxcnop 22436 Extend class notation with the operator norm function.

Syntaxccop 22437 Extend class notation with set of continuous Hilbert space operators.

Syntaxclo 22438 Extend class notation with set of linear Hilbert space operators.

Syntaxcbo 22439 Extend class notation with set of bounded linear operators.

Syntaxcuo 22440 Extend class notation with set of unitary Hilbert space operators.

Syntaxcho 22441 Extend class notation with set of Hermitian Hilbert space operators.

Syntaxcnmf 22442 Extend class notation with the functional norm function.

Syntaxcnl 22443 Extend class notation with the functional nullspace function.

Syntaxccnfn 22444 Extend class notation with set of continuous Hilbert space functionals.

Syntaxclf 22445 Extend class notation with set of linear Hilbert space functionals.

Syntaxcado 22446 Extend class notation with Hilbert space adjoint function.

Syntaxcbr 22447 Extend class notation with the bra of a vector in Dirac bra-ket notation.

Syntaxck 22448 Extend class notation with the outer product of two vectors in Dirac bra-ket notation.

Syntaxcleo 22449 Extend class notation with positive operator ordering.

Syntaxcei 22450 Extend class notation with Hilbert space eigenvector function.

Syntaxcel 22451 Extend class notation with Hilbert space eigenvalue function.

Syntaxcspc 22452 Extend class notation with the spectrum of an operator.

Syntaxcst 22453 Extend class notation with set of states on a Hilbert lattice.

Syntaxchst 22454 Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.

Syntaxccv 22455 Extend class notation with the covers relation on a Hilbert lattice.

Syntaxcat 22456 Extend class notation with set of atoms on a Hilbert lattice.
HAtoms

Syntaxcmd 22457 Extend class notation with the modular pair relation on a Hilbert lattice.

Syntaxcdmd 22458 Extend class notation with the dual modular pair relation on a Hilbert lattice.

18.1.2  Preliminary ZFC lemmas

Definitiondf-hnorm 22459 Define the function for the norm of a vector of Hilbert space. See normval 22614 for its value and normcl 22615 for its closure. Theorems norm-i-i 22623, norm-ii-i 22627, and norm-iii-i 22629 show it has the expected properties of a norm. In the literature, the norm of is usually written "|| ||", but we use function notation to take advantage of our existing theorems about functions. Definition of norm in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Definitiondf-hba 22460 Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 22490). Note that is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. This definition can be proved independently from those axioms as theorem hhba 22657. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Definitiondf-h0v 22461 Define the zero vector of Hilbert space. Note that is considered a primitive in the Hilbert space axioms below, and we don't use this definition outside of this section. It is proved from the axioms as theorem hh0v 22658. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Definitiondf-hvsub 22462* Define vector subtraction. See hvsubvali 22511 for its value and hvsubcli 22512 for its closure. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Definitiondf-hlim 22463* Define the limit relation for Hilbert space. See hlimi 22678 for its relational expression. Note that is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of converge in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Definitiondf-hcau 22464* Define the set of Cauchy sequences on a Hilbert space. See hcau 22674 for its membership relation. Note that is an infinite sequence of vectors, i.e. a mapping from integers to vectors. Definition of Cauchy sequence in [Beran] p. 96. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Theoremh2hva 22465 The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremh2hsm 22466 The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremh2hnm 22467 The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
CV

Theoremh2hvs 22468 The vector subtraction operation of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremh2hmetdval 22469 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Theoremh2hcau 22470 The Cauchy sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)

Theoremh2hlm 22471 The limit sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)

18.1.3  Derive the Hilbert space axioms from ZFC set theory

Before introducing the 18 axioms for Hilbert space, we first prove them as the conclusions of theorems axhilex-zf 22472 through axhcompl-zf 22489, using ZFC set theory only. These show that if we are given a known, fixed Hilbert space that satisfies their hypotheses, then we can derive the Hilbert space axioms as theorems of ZFC set theory. In practice, in order to use these theorems to convert the Hilbert Space explorer to a ZFC-only subtheory, we would also have to provide definitions for the 3 (otherwise primitive) class constants , , and before df-hnorm 22459 above. See also the comment in ax-hilex 22490.

Theoremaxhilex-zf 22472 Derive axiom ax-hilex 22490 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhfvadd-zf 22473 Derive axiom ax-hfvadd 22491 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvcom-zf 22474 Derive axiom ax-hvcom 22492 from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008.) (New usage is discouraged.)

Theoremaxhvass-zf 22475 Derive axiom ax-hvass 22493 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhv0cl-zf 22476 Derive axiom ax-hv0cl 22494 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvaddid-zf 22477 Derive axiom ax-hvaddid 22495 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhfvmul-zf 22478 Derive axiom ax-hfvmul 22496 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvmulid-zf 22479 Derive axiom ax-hvmulid 22497 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvmulass-zf 22480 Derive axiom ax-hvmulass 22498 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvdistr1-zf 22481 Derive axiom ax-hvdistr1 22499 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvdistr2-zf 22482 Derive axiom ax-hvdistr2 22500 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvmul0-zf 22483 Derive axiom ax-hvmul0 22501 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhfi-zf 22484 Derive axiom ax-hfi 22569 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhis1-zf 22485 Derive axiom ax-his1 22572 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhis2-zf 22486 Derive axiom ax-his2 22573 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhis3-zf 22487 Derive axiom ax-his3 22574 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhis4-zf 22488 Derive axiom ax-his4 22575 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhcompl-zf 22489* Derive axiom ax-hcompl 22692 from Hilbert space under ZF set theory. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)

18.1.4  Introduce the vector space axioms for a Hilbert space

Here we introduce the axioms a complex Hilbert space, which is the foundation for quantum mechanics and quantum field theory. The 18 axioms for a complex Hilbert space consist of ax-hilex 22490, ax-hfvadd 22491, ax-hvcom 22492, ax-hvass 22493, ax-hv0cl 22494, ax-hvaddid 22495, ax-hfvmul 22496, ax-hvmulid 22497, ax-hvmulass 22498, ax-hvdistr1 22499, ax-hvdistr2 22500, ax-hvmul0 22501, ax-hfi 22569, ax-his1 22572, ax-his2 22573, ax-his3 22574, ax-his4 22575, and ax-hcompl 22692.

The axioms specify the properties of 5 primitive symbols, , , , , and .

If we can prove in ZFC set theory that a class is a complex Hilbert space, i.e. that , then these axioms can be proved as theorems axhilex-zf 22472, axhfvadd-zf 22473, axhvcom-zf 22474, axhvass-zf 22475, axhv0cl-zf 22476, axhvaddid-zf 22477, axhfvmul-zf 22478, axhvmulid-zf 22479, axhvmulass-zf 22480, axhvdistr1-zf 22481, axhvdistr2-zf 22482, axhvmul0-zf 22483, axhfi-zf 22484, axhis1-zf 22485, axhis2-zf 22486, axhis3-zf 22487, axhis4-zf 22488, and axhcompl-zf 22489 respectively. In that case, the theorems of the Hilbert Space Explorer will become theorems of ZFC set theory. See also the comments in axhilex-zf 22472.

Axiomax-hilex 22490 This is our first axiom for a complex Hilbert space, which is the foundation for quantum mechanics and quantum field theory. We assume that there exists a primitive class, , which contains objects called vectors. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Axiomax-hfvadd 22491 Vector addition is an operation on . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Axiomax-hvcom 22492 Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Axiomax-hvass 22493 Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Axiomax-hv0cl 22494 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)

Axiomax-hvaddid 22495 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Axiomax-hfvmul 22496 Scalar multiplication is an operation on and . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Axiomax-hvmulid 22497 Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Axiomax-hvmulass 22498 Scalar multiplication associative law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Axiomax-hvdistr1 22499 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Axiomax-hvdistr2 22500 Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

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