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Theorem List for Metamath Proof Explorer - 21301-21400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhlnvi 21301 Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Theoremhlvc 21302 Every complex Hilbert space is a complex vector space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlcmet 21303 The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlmet 21304 The induced metric on a complex Hilbert space. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlpar2 21305 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
CV

Theoremhlpar 21306 The parallelogram law satified by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)
CV

15.8.2  Standard axioms for a complex Hilbert space

Theoremhlex 21307 The base set of a Hilbert space is a set. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhladdf 21308 Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlcom 21309 Hilbert space vector addition is commutative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlass 21310 Hilbert space vector addition is associative. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhl0cl 21311 The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhladdid 21312 Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmulf 21313 Mapping for Hilbert space scalar multiplication. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmulid 21314 Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmulass 21315 Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhldi 21316 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhldir 21317 Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlmul0 21318 Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Theoremhlipf 21319 Mapping for Hilbert space inner product. (Contributed by NM, 19-Nov-2007.) (New usage is discouraged.)

Theoremhlipcj 21320 Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipdir 21321 Distributive law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipass 21322 Associative law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlipgt0 21323 The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.)

Theoremhlcompl 21324 Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007.) (Revised by Mario Carneiro, 9-May-2014.) (New usage is discouraged.)

15.8.3  Examples of complex Hilbert spaces

Theoremcnchl 21325 The set of complex numbers is a complex Hilbert space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.)

15.8.4  Subspaces

Theoremssphl 21326 A Banach subspace of an inner product space is a Hilbert space. (Contributed by NM, 11-Apr-2008.) (New usage is discouraged.)

15.8.5  Hellinger-Toeplitz Theorem

Theoremhtthlem 21327* Lemma for htth 21328. The collection , which consists of functions for each in the unit ball, is a collection of bounded linear functions by ipblnfi 21264, so by the Uniform Boundedness theorem ubth 21282, 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 21328* 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.)

15.9  Hilbert Space Explorer

15.9.1  Basic Hilbert space definitions

Syntaxchil 21329 Extend class notation with Hilbert vector space.

Syntaxcva 21330 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 8620.

Syntaxcsm 21331 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 21332 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 3553.

Syntaxcno 21333 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 21334 Extend class notation with zero vector in Hilbert space.

Syntaxcmv 21335 Extend class notation with vector subtraction in Hilbert space.

Syntaxccau 21336 Extend class notation with set of Cauchy sequences in Hilbert space.

Syntaxchli 21337 Extend class notation with convergence relation in Hilbert space.

Syntaxcsh 21338 Extend class notation with set of subspaces of a Hilbert space.

Syntaxcch 21339 Extend class notation with set of closed subspaces of a Hilbert space.

Syntaxcort 21340 Extend class notation with orthogonal complement in .

Syntaxcph 21341 Extend class notation with subspace sum in .

Syntaxcspn 21342 Extend class notation with subspace span in .

Syntaxchj 21343 Extend class notation with join in .

Syntaxchsup 21344 Extend class notation with supremum of a collection in .

Syntaxc0h 21345 Extend class notation with zero of .

Syntaxccm 21346 Extend class notation with the commutes relation on a Hilbert lattice.

Syntaxcpjh 21347 Extend class notation with set of projections on a Hilbert space.

Syntaxchos 21348 Extend class notation with sum of Hilbert space operators.

Syntaxchot 21349 Extend class notation with scalar product of a Hilbert space operator.

Syntaxchod 21350 Extend class notation with difference of Hilbert space operators.

Syntaxchfs 21351 Extend class notation with sum of Hilbert space functionals.

Syntaxchft 21352 Extend class notation with scalar product of Hilbert space functional.

Syntaxch0o 21353 Extend class notation with the Hilbert space zero operator.

Syntaxchio 21354 Extend class notation with Hilbert space identity operator.

Syntaxcnop 21355 Extend class notation with the operator norm function.

Syntaxccop 21356 Extend class notation with set of continuous Hilbert space operators.

Syntaxclo 21357 Extend class notation with set of linear Hilbert space operators.

Syntaxcbo 21358 Extend class notation with set of bounded linear operators.

Syntaxcuo 21359 Extend class notation with set of unitary Hilbert space operators.

Syntaxcho 21360 Extend class notation with set of Hermitian Hilbert space operators.

Syntaxcnmf 21361 Extend class notation with the functional norm function.

Syntaxcnl 21362 Extend class notation with the functional nullspace function.

Syntaxccnfn 21363 Extend class notation with set of continuous Hilbert space functionals.

Syntaxclf 21364 Extend class notation with set of linear Hilbert space functionals.

Syntaxcbr 21366 Extend class notation with the bra of a vector in Dirac bra-ket notation.

Syntaxck 21367 Extend class notation with the outer product of two vectors in Dirac bra-ket notation.

Syntaxcleo 21368 Extend class notation with positive operator ordering.

Syntaxcei 21369 Extend class notation with Hilbert space eigenvector function.

Syntaxcel 21370 Extend class notation with Hilbert space eigenvalue function.

Syntaxcspc 21371 Extend class notation with the spectrum of an operator.

Syntaxcst 21372 Extend class notation with set of states on a Hilbert lattice.

Syntaxchst 21373 Extend class notation with set of Hilbert-space-valued states on a Hilbert lattice.

Syntaxccv 21374 Extend class notation with the covers relation on a Hilbert lattice.

Syntaxcat 21375 Extend class notation with set of atoms on a Hilbert lattice.
HAtoms

Syntaxcmd 21376 Extend class notation with the modular pair relation on a Hilbert lattice.

Syntaxcdmd 21377 Extend class notation with the dual modular pair relation on a Hilbert lattice.

15.9.2  Preliminary ZFC lemmas

Definitiondf-hnorm 21378 Define the function for the norm of a vector of Hilbert space. See normval 21533 for its value and normcl 21534 for its closure. Theorems norm-i-i 21542, norm-ii-i 21546, and norm-iii-i 21548 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 21379 Define base set of Hilbert space, for use if we want to develop Hilbert space independently from the axioms (see comments in ax-hilex 21409). 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 21576. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Definitiondf-h0v 21380 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 21577. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Definitiondf-hvsub 21381* Define vector subtraction. See hvsubvali 21430 for its value and hvsubcli 21431 for its closure. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Definitiondf-hlim 21382* Define the limit relation for Hilbert space. See hlimi 21597 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 21383* Define the set of Cauchy sequences on a Hilbert space. See hcau 21593 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 21384 The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremh2hsm 21385 The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremh2hnm 21386 The norm function of Hilbert space. (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)
CV

Theoremh2hvs 21387 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 21388 Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)

Theoremh2hcau 21389 The Cauchy sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)

Theoremh2hlm 21390 The limit sequences of Hilbert space. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 13-May-2014.) (New usage is discouraged.)

15.9.3  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 21391 through axhcompl-zf 21408, 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 21378 above. See also the comment in ax-hilex 21409.

Theoremaxhilex-zf 21391 Derive axiom ax-hilex 21409 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhfvadd-zf 21392 Derive axiom ax-hfvadd 21410 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvcom-zf 21393 Derive axiom ax-hvcom 21411 from Hilbert space under ZF set theory. (Contributed by NM, 27-May-2008.) (New usage is discouraged.)

Theoremaxhvass-zf 21394 Derive axiom ax-hvass 21412 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhv0cl-zf 21395 Derive axiom ax-hv0cl 21413 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvaddid-zf 21396 Derive axiom ax-hvaddid 21414 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhfvmul-zf 21397 Derive axiom ax-hfvmul 21415 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvmulid-zf 21398 Derive axiom ax-hvmulid 21416 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvmulass-zf 21399 Derive axiom ax-hvmulass 21417 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

Theoremaxhvdistr1-zf 21400 Derive axiom ax-hvdistr1 21418 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)

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