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Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcsm 21501 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 21502 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 3649.

Syntaxcno 21503 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 21504 Extend class notation with zero vector in Hilbert space.

Syntaxcmv 21505 Extend class notation with vector subtraction in Hilbert space.

Syntaxccau 21506 Extend class notation with set of Cauchy sequences in Hilbert space.

Syntaxchli 21507 Extend class notation with convergence relation in Hilbert space.

Syntaxcsh 21508 Extend class notation with set of subspaces of a Hilbert space.

Syntaxcch 21509 Extend class notation with set of closed subspaces of a Hilbert space.

Syntaxcort 21510 Extend class notation with orthogonal complement in .

Syntaxcph 21511 Extend class notation with subspace sum in .

Syntaxcspn 21512 Extend class notation with subspace span in .

Syntaxchj 21513 Extend class notation with join in .

Syntaxchsup 21514 Extend class notation with supremum of a collection in .

Syntaxc0h 21515 Extend class notation with zero of .

Syntaxccm 21516 Extend class notation with the commutes relation on a Hilbert lattice.

Syntaxcpjh 21517 Extend class notation with set of projections on a Hilbert space.

Syntaxchos 21518 Extend class notation with sum of Hilbert space operators.

Syntaxchot 21519 Extend class notation with scalar product of a Hilbert space operator.

Syntaxchod 21520 Extend class notation with difference of Hilbert space operators.

Syntaxchfs 21521 Extend class notation with sum of Hilbert space functionals.

Syntaxchft 21522 Extend class notation with scalar product of Hilbert space functional.

Syntaxch0o 21523 Extend class notation with the Hilbert space zero operator.

Syntaxchio 21524 Extend class notation with Hilbert space identity operator.

Syntaxcnop 21525 Extend class notation with the operator norm function.

Syntaxccop 21526 Extend class notation with set of continuous Hilbert space operators.

Syntaxclo 21527 Extend class notation with set of linear Hilbert space operators.

Syntaxcbo 21528 Extend class notation with set of bounded linear operators.

Syntaxcuo 21529 Extend class notation with set of unitary Hilbert space operators.

Syntaxcho 21530 Extend class notation with set of Hermitian Hilbert space operators.

Syntaxcnmf 21531 Extend class notation with the functional norm function.

Syntaxcnl 21532 Extend class notation with the functional nullspace function.

Syntaxccnfn 21533 Extend class notation with set of continuous Hilbert space functionals.

Syntaxclf 21534 Extend class notation with set of linear Hilbert space functionals.

Syntaxcbr 21536 Extend class notation with the bra of a vector in Dirac bra-ket notation.

Syntaxck 21537 Extend class notation with the outer product of two vectors in Dirac bra-ket notation.

Syntaxcleo 21538 Extend class notation with positive operator ordering.

Syntaxcei 21539 Extend class notation with Hilbert space eigenvector function.

Syntaxcel 21540 Extend class notation with Hilbert space eigenvalue function.

Syntaxcspc 21541 Extend class notation with the spectrum of an operator.

Syntaxcst 21542 Extend class notation with set of states on a Hilbert lattice.

Syntaxchst 21543 Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.

Syntaxccv 21544 Extend class notation with the covers relation on a Hilbert lattice.

Syntaxcat 21545 Extend class notation with set of atoms on a Hilbert lattice.
HAtoms

Syntaxcmd 21546 Extend class notation with the modular pair relation on a Hilbert lattice.

Syntaxcdmd 21547 Extend class notation with the dual modular pair relation on a Hilbert lattice.

17.1.2  Preliminary ZFC lemmas

Definitiondf-hnorm 21548 Define the function for the norm of a vector of Hilbert space. See normval 21703 for its value and normcl 21704 for its closure. Theorems norm-i-i 21712, norm-ii-i 21716, and norm-iii-i 21718 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 21549 Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 21579). 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 as theorem hhba 21746. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Definitiondf-h0v 21550 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 21747. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Definitiondf-hvsub 21551* Define vector subtraction. See hvsubvali 21600 for its value and hvsubcli 21601 for its closure. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Definitiondf-hlim 21552* Define the limit relation for Hilbert space. See hlimi 21767 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 21553* Define the set of Cauchy sequences on a Hilbert space. See hcau 21763 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 21554 The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremh2hsm 21555 The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremh2hnm 21556 The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
CV

Theoremh2hvs 21557 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 21558 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Theoremh2hcau 21559 The Cauchy sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)

Theoremh2hlm 21560 The limit sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)

17.1.3  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 21561 through axhcompl-zf 21578, 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 21548 above. See also the comment in ax-hilex 21579.

Theoremaxhilex-zf 21561 Derive axiom ax-hilex 21579 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhfvadd-zf 21562 Derive axiom ax-hfvadd 21580 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvcom-zf 21563 Derive axiom ax-hvcom 21581 from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008.) (New usage is discouraged.)

Theoremaxhvass-zf 21564 Derive axiom ax-hvass 21582 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhv0cl-zf 21565 Derive axiom ax-hv0cl 21583 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvaddid-zf 21566 Derive axiom ax-hvaddid 21584 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhfvmul-zf 21567 Derive axiom ax-hfvmul 21585 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvmulid-zf 21568 Derive axiom ax-hvmulid 21586 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvmulass-zf 21569 Derive axiom ax-hvmulass 21587 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvdistr1-zf 21570 Derive axiom ax-hvdistr1 21588 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvdistr2-zf 21571 Derive axiom ax-hvdistr2 21589 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvmul0-zf 21572 Derive axiom ax-hvmul0 21590 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhfi-zf 21573 Derive axiom ax-hfi 21658 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhis1-zf 21574 Derive axiom ax-his1 21661 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhis2-zf 21575 Derive axiom ax-his2 21662 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhis3-zf 21576 Derive axiom ax-his3 21663 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhis4-zf 21577 Derive axiom ax-his4 21664 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhcompl-zf 21578* Derive axiom ax-hcompl 21781 from Hilbert space under ZF set theory. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)

17.1.4  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 21579, ax-hfvadd 21580, ax-hvcom 21581, ax-hvass 21582, ax-hv0cl 21583, ax-hvaddid 21584, ax-hfvmul 21585, ax-hvmulid 21586, ax-hvmulass 21587, ax-hvdistr1 21588, ax-hvdistr2 21589, ax-hvmul0 21590, ax-hfi 21658, ax-his1 21661, ax-his2 21662, ax-his3 21663, ax-his4 21664, and ax-hcompl 21781.

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 21561, axhfvadd-zf 21562, axhvcom-zf 21563, axhvass-zf 21564, axhv0cl-zf 21565, axhvaddid-zf 21566, axhfvmul-zf 21567, axhvmulid-zf 21568, axhvmulass-zf 21569, axhvdistr1-zf 21570, axhvdistr2-zf 21571, axhvmul0-zf 21572, axhfi-zf 21573, axhis1-zf 21574, axhis2-zf 21575, axhis3-zf 21576, axhis4-zf 21577, and axhcompl-zf 21578 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 21561.

Axiomax-hilex 21579 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 21580 Vector addition is an operation on . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Axiomax-hvcom 21581 Vector addition is commutative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Axiomax-hvass 21582 Vector addition is associative. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Axiomax-hv0cl 21583 The zero vector is in the vector space. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)

Axiomax-hvaddid 21584 Addition with the zero vector. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Axiomax-hfvmul 21585 Scalar multiplication is an operation on and . (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

Axiomax-hvmulid 21586 Scalar multiplication by one. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Axiomax-hvmulass 21587 Scalar multiplication associative law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Axiomax-hvdistr1 21588 Scalar multiplication distributive law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Axiomax-hvdistr2 21589 Scalar multiplication distributive law. (Contributed by NM, 30-May-1999.) (New usage is discouraged.)

Axiomax-hvmul0 21590 Scalar multiplication by zero. We can derive the existence of the negative of a vector from this axiom (see hvsubid 21605 and hvsubval 21596). (Contributed by NM, 29-May-1999.) (New usage is discouraged.)

17.1.5  Vector operations

Theoremhvmulex 21591 The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.)

Theoremhvaddcl 21592 Closure of vector addition. (Contributed by NM, 18-Apr-2007.) (New usage is discouraged.)

Theoremhvmulcl 21593 Closure of scalar multiplication. (Contributed by NM, 19-Apr-2007.) (New usage is discouraged.)

Theoremhvmulcli 21594 Closure inference for scalar multiplication. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)

Theoremhvsubf 21595 Mapping domain and codomain of vector subtraction. (Contributed by NM, 6-Sep-2007.) (New usage is discouraged.)

Theoremhvsubval 21596 Value of vector subtraction. (Contributed by NM, 5-Sep-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Theoremhvsubcl 21597 Closure of vector subtraction. (Contributed by NM, 17-Aug-1999.) (New usage is discouraged.)

Theoremhvaddcli 21598 Closure of vector addition. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)

Theoremhvcomi 21599 Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Theoremhvsubvali 21600 Value of vector subtraction definition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

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